BROWNIAN BROWNIAN MOTION – I
N. CHERNOV AND D. DOLGOPYAT
1. Introduction
1.1. The model. Let B1 , . . . , Br be open convex domains on the unit
two dimensional torus T2 . Assume that B̄i ∩ B̄j = ∅ for i 6= j, and
for each i the boundary ∂Bi is a C 3 smooth closed curve with a nonvanishing curvature.
We study a dynamical system of two particles – a hard disk of mass
M 1 and radius r > 0 and a light point particle of mass 1. These
particles move freely inside D := T2 \ ∪i Bi and collide elastically with
each other and with the fixed obstacles Bi . We call Bi scatterers.
Let Q(t) denote the center and V (t) the velocity of the heavy disk at
time t. Similarly, let q(t) denote the position of the light particle and
v(t) its velocity. When a particle collides with a scatterer, the normal
component of its velocity reverses. When the two particles collide with
each other, the normal components of their velocities change by the
rules
M −1 ⊥
2M
⊥
(1.1)
vnew
=−
vold +
V⊥
M +1
M + 1 old
and
2
M −1 ⊥
⊥
⊥
Vold +
vold
,
(1.2)
Vnew
=
M +1
M +1
while the tangential components remain unchanged. The total kinetic
energy is preserved, and we fix it so that
kvk2 + M kV k2 = 1.
√
This implies kvk ≤ 1 and kV k ≤ 1/ M .
Our system is Hamiltonian and preserves a Liouville measure. Systems of hard disks in closed containers are proven to be completely
hyperbolic and ergodic under various conditions [10, 50, 51]. These results do not cover our particular model, but we have little doubt that it
is hyperbolic and ergodic, too. In this paper, though, we do not study
ergodic properties.
We are interested in the evolution of the system during the initial period of time before the heavy disk experiences its first collision with the
(1.3)
1
2
N. CHERNOV AND D. DOLGOPYAT
border ∂D. This restricts our analysis to an interval of time (0, cM a ),
where c, a > 0 depend on Q(0) and V (0), see below. During this initial
period, the system does not exhibit its ergodic behavior, but it does
exhibit a diffusive behavior
in the following sense. As (1.1)–(1.2) imply,
√
kvnew k−kvold k ≤ 2/ M and kVnew −Vold k ≤ 2/M , hence the changes in
kvk and V at each collision are much smaller
√ than their typical values,
which are kvk = O(1) and kV k = O(1/ M ). Thus, the speed of the
light particle, kv(t)k, remains almost constant, and the heavy particle
not only moves slowly but its velocity V (t) changes slowly as well (it
has inertia). We will show that, in the limit M → ∞, the velocity V (t)
can be approximated by a Brownian motion, and the position Q(t) by
an integral of the Brownian motion.
Our system is one of the simplest models of a particle moving in a
fluid. This is what scientists called Brownian motion about one hundred years ago. Now this term has a more narrow technical meaning,
namely a Gaussian process with zero mean and stationary independent
increments. Our paper is motivated by Brownian motion in its original
sense, and we call it “Brownian Brownian motion”.
Even though this paper only covers two particle systems (where the
“fluid” is represented by a single light particle), we believe that our
methods can extend to more realistic fluids of many particles. We
consider this paper as a first step in our studies (thus the number one
in its title) and plan to investigate more complex models in the future,
see our discussion in Section 7.
1.2. Billiard approximations. We denote phase points by x = (Q, V, q, v)
and the phase space by M. Due to the energy conservation (1.3),
dim M = 7. The dynamics Φt : M → M can be reduced, in a standard way, to a discrete time collision map as follows.
We call Ω = ∂M the collision space. Let P(Q) denote the disk of radius r centered on Q, then Ω = {(Q, V, q, v) ∈ M : q ∈ ∂D ∪ ∂P(Q)}.
At each collision, we identify the precollisional and postcollisional velocity vectors. Technically, we will only include the postcollisional vector
in Ω, so that
Ω = Ω D ∪ ΩP ,
ΩD = {(Q, V, q, v) ∈ M : q ∈ ∂D, hv, ni ≥ 0},
ΩP = {(Q, V, q, v) ∈ M : q ∈ ∂P(Q), hv − V, ni ≥ 0}.
where h·, ·i stands for the scalar product of vectors and n denotes a
normal vector to ∂D ∪ ∂P (Q) at q pointing into D \ P(Q). The first
return map F : Ω → Ω is called the collision map. It preserves a
BROWNIAN BROWNIAN MOTION
3
smooth probability measure µ on Ω induced by the Liouville measure
on M.
It will be convenient to denote points of Ω by (Q, V, q, w), where
v
for q ∈ ∂D
(1.4)
w=
v − V for q ∈ ∂P(Q)
so that
(1.5)
Ω = {(Q, V, q, w) : q ∈ ∂D ∪ ∂P(Q), hw, ni ≥ 0}
For every point (Q, V, q, w) ∈ Ω we put
p
w
(1.6)
w̄ =
SV , SV = 1 − M kV k2
kwk
For each pair (Q, V ) we denote by ΩQ,V the cross-section of Ω obtained
by fixing Q and V . By using the w̄ notation, we can write
(1.7) ΩQ,V = {(q, w̄) : q ∈ ∂D ∪ ∂P(Q), hw̄, ni ≥ 0, kw̄k = SV }
Now let us pick t0 ≥ 0 and fix the center of the heavy disk at
Q = Q(t0 ) ∈ D and set M = ∞. Then the light particle would
move with a constant speed kv(t)k = SV , where V = V (t0 ), in the
domain D \ P(Q) with specular reflections at ∂D ∪ ∂P (Q). Thus we
get a billiard-type dynamics, which approximates our system during a
relatively short interval of time, until our heavy disk moves a considerable distance. We may treat our system then as a small perturbation
of this billiard-type dynamics, and in fact our entire analysis is based
on this approximation.
The collision map FQ,V of the above billiard system acts on the
space (1.7), where w̄ denotes the postcollisional velocity of the moving
particle. The map FQ,V : ΩQ,V → ΩQ,V preserves a smooth probability
measure µQ,V as described in Section 3.
The billiard-type system (ΩQ,V , FQ,V , µQ,V ) is essentially independent of V . By a simple rescaling (i.e. renormalizing) of w̄ we can
identify it with (ΩQ,0 , FQ,0 , µQ,0 ), which we denote, for brevity, by
(ΩQ , FQ , µQ ), and it becomes a standard billiard system (where the
particle moves at unit speed) on the table D \ ∂P (Q). This is a dispersing (Sinai) table, hence the map FQ is hyperbolic, ergodic and has
strong statistical properties, including exponential decay of correlations
and the central limit theorem [53, 57].
Equations (1.1)–(1.2) imply that the change of the velocity of the
disk due to a collision with the light particle is
⊥
⊥
2 vnew
− Vnew
2 w⊥
=−
(1.8)
Vnew − Vold = −
M +1
M +1
4
N. CHERNOV AND D. DOLGOPYAT
Since w = w̄ kv − V k/kvk, we have
(1.9)
Vnew − Vold = −
2 w̄ ⊥
+χ
M
where
(1.10)
|χ| ≤ Const
kV k
1
+ 2
M kvk M
is a relatively small term. Define a vector function on Ω by
−2w̄ ⊥ for q ∈ ∂P (Q)
(1.11)
A=
0
for q ∈ ∂D \ ∂P(Q)
RObviously, A is a smooth function, and due to a rotational symmetry
A dµQ,V = 0 for every Q, V . Hence the central limit theorem [7, 57]
implies the convergence in distribution
n−1
(1.12)
1 X
j
2
√
(A)).
A ◦ FQ,V
→ N (0, σ̄Q,V
n j=0
2
as n → ∞, where σ̄Q,V
(A) a positive semidefinite matrix given by the
Green-Kubo formula
∞ Z
X
T
j
2
A A ◦ FQ,V
dµQ,V .
(1.13)
σ̄Q,V (A) =
j=−∞
(this series converges because its terms decay exponentially fast as
2
(A) :=
|j| → ∞, see [57]). By setting V = 0 we define a matrix σ̄Q
2
σ̄Q,0 (A). Since the restriction of A to the sets ΩQ,V and ΩQ,0 = ΩQ
only differ by a scaling factor SV , see (1.6), we have a simple relation
(1.14)
2
2
σ̄Q,V
(A) = (1 − M kV k2 ) σ̄Q
(A)
2
We prove in Appendix A that the matrix σ̄Q
(A) depends continuously
2
on Q and, for typical domains D, is nonsingular, i.e. σ̄Q
(A) > 0. Let
2
2
σQ
(A) = σ̄Q
(A)/l, where
(1.15)
l=π
Area(D) − Area(P)
length(∂D) + length(∂P )
is the mean free path of the light particle in the billiard dynamics FQ ,
see [14] (observe that l does not depend on Q). Lastly, let σQ (A) be
2
the positive semidefinite square root of σQ
(A).
Let us draw some conclusions, which are entirely heuristic at this
point and will be formalized later. In view of (1.8)–(1.12), we may
expect that the total change of the disk velocity V due to n consecutive
collisions of the light particle with ∂D ∪ ∂P can be approximated by
BROWNIAN BROWNIAN MOTION
5
2
a normal random variable N 0, nσ̄Q,V
(A)/M 2 . During an interval
(t0 , t1 ), the light particle experiences n ≈ kvk(t1 −t0 )/l collisions, hence
the total change of the disk velocity may be approximated by
2
(1.16)
V (t1 ) − V (t0 ) ∼ N 0, kvk(t1 − t0 ) σQ,V
(A)/M 2
(assuming that Q(t) ≈ Q(t0 ) and kv(t)k ≈ SV (t0 ) for all t0 < t < t1 ).
A large part of our paper is devoted to making the approximation (1.16)
precise.
1.3. Statement of results. We always assume that the domain D
has finite horizon, i.e. the light particle cannot travel a distance > 1
without collisions, even if we remove the massive disk from D.
Suppose the initial position Q(0) = Q0 and velocity V (0) = V0 of
the heavy particle are fixed, and the initial state of the light particle
q(0), v(0) is selected randomly, according to a smooth distribution in
the direct product of the domain D \ P(Q0 ) and the circle kv(0)k2 =
1 − M kV0 k2 (alternatively, q(0) may be chosen ∂D ∪ ∂P (Q0 )). The
shape of the initial distribution will not affect our results.
We consider the trajectory of the heavy particle Q(t), V (t) during
a time interval (0, cM a ) with some c, a > 0 selected below. We scale
time by τ = t/M a and, sometimes, scale space in a way specified below,
to convert Q(t), V (t) to a pair of functions Q(τ ), V(τ ) on the interval
0 < τ < c. The random choice of q(0), v(0) induces a probability
measure on the space of functions Q(τ ), V(τ ), and we are interested in
the convergence of this probability measure, as M → ∞, to a stochastic
process Q(τ ), V(τ ). We prove three results in this direction.
√
First, let the initial velocity of the heavy particle be of order 1/ M .
Specifically, let us fix Q0 and a unit vector u0 and set V0 = u0 ηM −1/2
for some η ∈ (0, 1). Note that if the heavy disk moved with a constant
velocity, V0 , without colliding with the light particle, it would hit ∂D
at a certain moment c0 M 1/2 , where c0 > 0 is determined by Q0 , u0
and η. We restrict our analysis to a time interval (0, cM 1/2 ) with some
c < c0 . During this period of time we expect, due to (1.16), that the
overall fluctuations of the disk velocity will be O(M −3/4 ). Hence we
expect V (t) = V0 + O(M −3/4 ) and Q(t) = Q0 + tV0 + O(M −1/4 ) for
0 < t < cM 1/2 . This leads us to a time scale τ = tM −1/2 and a space
scale
Q(τ ) = M 1/4 Q(τ M 1/2 ) − Q0 − τ M 1/2 V0
and, respectively,
V(τ ) = M 3/4 V (τ M 1/2 ) − V0
6
N. CHERNOV AND D. DOLGOPYAT
We can find an asymptotic distribution of V(τ ) by using our heuristic
normal approximation (1.16). Let
Q† (τ ) = Q0 + τ M 1/2 V0 = Q0 + τ ηu0
Then for any τ ∈ (0, c) we have Q(τ M 1/2 ) → Q† (τ ), as M → ∞, hence
2
2
σQ(τ
M 1/2 ) (A) → σQ† (τ ) (A).
p
Anticipating that kv(t)k ≈
1 − η 2 for all 0 < t < cM 1/2 we can
expect that for small
pdτ > 0 the number of collisions N (dτ ) is approximately equal to dτ p1 − η 2 /l and the momenta exchange during each
collision is close to 1 − η 2 A. This would give
V(τ + dτ ) − V(τ ) → N 0, D(dτ )
(1.17)
where
(1.18)
2
D(dτ ) ≈ N (dτ )(1 − η 2 )σ̄Q
† (τ ) (A) = dτ
Integrating (1.18) over τ yields
Z
p
2
3
(1.19)
V(τ ) → N 0, (1 − η )
p
2
(1 − η 2 )3 σQ
† (τ ) (A)
τ
0
2
σQ
† (s) (A) ds
The following theorem makes this conclusion precise:
Theorem 1. Under the above assumptions, the function V(τ ) for 0 <
τ < c converges weakly to a Gaussian Markov random process V(τ )
with independent increments, zero mean and covariance matrix
Z τ
p
2
2
3
(1.20)
Cov(V(τ )) = (1 − η )
σQ
† (s) (A) ds
0
The process V(τ ) can be, equivalently, defined by
Z τ
p
4
2
3
(1.21)
V(τ ) = (1 − η )
σQ† (s) (A) dw(s)
0
where w(s) denotes the standard two dimensional Brownian motion.
Accordingly, theRfunction Q(τ ) converges weakly to a Gaussian random
τ
process Q(τ ) = 0 V(s) ds, which has zero mean and covariance matrix
Z τ
p
2
2
3
(τ − s)2 σQ
(1.22)
Cov(Q(τ )) = (1 − η )
† (s) (A) ds
0
We note that the limit velocity process is a (time inhomogeneous)
Brownian motion, while the limit position process is the integral of
that Brownian motion.
The next theorem deals with a more difficult case V0 = 0. Since the
heavy disk is now initially at rest, it takes it longer to build up speed
BROWNIAN BROWNIAN MOTION
7
and move to the border ∂D. We
√ expect, due to (1.16), that the disk
velocity grows as kV (t)k = O( t/M ), and therefore its displacement
grows as kQ(t) − Q0 k = O(t3/2 /M ). Hence, for typical trajectories, it
takes O(M 2/3 ) units of time before the disk hits ∂D. It is convenient to
modify the dynamics of the disk making it stop (“freeze”) the moment
it comes too close to ∂D. We pick a small δ0 > 0 and stop the disk at
the moment
(1.23)
t∗ = min{t > 0 : dist(Q(t), ∂D) = r + δ0 },
hence we obtain a modified dynamics Q∗ (t), V∗ (t) given by
Q(t) for t < t∗
V (t) for t < t∗
Q∗ (t) =
V∗ (t) =
.
Q(t∗ ) for t > t∗
0
for t > t∗
With this modification, we can consider the dynamics on a time interval
(0, cM 2/3 ) with an arbitrary c > 0. Our time scale is τ = tM −2/3 and
there is no need for the space scale, i.e. we set Q(τ ) = Q∗ (τ M 2/3 ) and
V(τ ) = M 2/3 V∗ (τ M 2/3 ).
We note that the heavy disk, being initially at rest, can move randomly in any direction, and follow a random trajectory before reaching
2
∂D. This leads to another complication. The matrix σQ(τ
(A) does
M 2/3 )
not have a limit, as M → ∞, for any τ > 0, because it depends on the
(random) location of the disk. Hence, heuristic estimates of the sort
(1.17)–(1.19) are now impossible. Due to these complications, we only
obtain a partial convergence result:
Theorem 2. Under the above assumptions, for each sequence Mk →
∞ there is a subsequence Mkj → ∞ along which the functions Q(τ )
and V(τ ) on the interval 0 < τ < c weakly converge to stochastic
processes Q(τ ) and V(τ ), respectively. These processes are solutions
of the following stochastic differential equations
dQ = V dτ,
(1.24)
dV = σQ (A) dw(τ ),
Q 0 = Q0
V0 = 0
(here again w(τ ) is the standard two dimensional Brownian motion)
which are stopped the moment Q(τ ) comes to within the distance r + δ0
from the border ∂D.
Stochastic differential equations (1.24) may have multiple solutions
2
if the matrix σQ
(A), as a function of Q, is not smooth enough, and we
2
can only prove that σQ
(A) is continuous. However, we conjecture that
2
(A) is sufficiently smooth to guarantee
in our model the function σQ
the uniqueness of the solution of (1.24) and hence the convergence, as
8
N. CHERNOV AND D. DOLGOPYAT
M → ∞, of Q(τ ), V(τ ) to Q(τ ), V(τ ) in the above theorem. We hope
to investigate this issue in a future publication.
The complication in the previous theorem can be removed by taking
the limit r → 0, in addition to M → ∞. In this case the matrix
2
σQ
(A) will be asymptotically constant, as we explain next. Recall that
2
2
2
σQ (A) = σ̄Q
(A)/l, where l does not depend on Q. Next, σ̄Q
(A) is given
by the Green-Kubo formula (1.13). Its central term, corresponding to
j = 0, can be found by a direct calculation:
Z
8πr
(1.25)
A AT dµQ =
I,
3 (length(∂D) + length(∂P))
2
and it is independent of Q. Hence, the dependence of σQ
(A) on Q only
comes from the correlation terms j 6= 0 in the series (1.13). When the
size of the massive disk is comparable to the size of the domain D, the
average time between its successive collisions with the light particle is
of order one, and those collisions are strongly correlated. By contrast,
if r ≈ 0, the average time between successive interparticle collisions is
O(1/r), and these collisions become almost independent. Thus, in the
Green-Kubo formula (1.13), the central term (1.25) becomes dominant,
and we arrive at
8r
2
(1.26)
σQ
(A) =
I + o(r).
3 Area(D)
see Appendix A for a complete proof.
Next, the time scale introduced in the previous theorem has to be
adjusted to the present case where r → 0. Due to (1.16)
√ and (1.26),
we expect that the disk velocity grows as kV (t)k = O( rt/M ), and
its displacement as kQ(t) − Q0 k = O(t3/2 r1/2 /M ). Hence, for typical
trajectories, it takes O(r−1/3 M 2/3 ) units of time before the heavy disk
hits ∂D. It is important that during this period of time kV (t)k =
O(r 1/3 M −2/3 ) ≈ 0, hence kvk remains close to one. We again modify
the dynamics of the disk making it stop (freeze up) the moment it
comes δ0 close to ∂D and consider the modified dynamics of the heavy
disk Q∗ (t), V∗ (t) on a time interval (0, c r−1/3 M 2/3 ) with a constant
c > 0. Our time scale is now τ = t r1/3 M −2/3 , and we set Q(τ ) =
Q∗ (τ r−1/3 M 2/3 ) and V(τ ) = r−1/3 M 2/3 V∗ (τ r−1/3 M 2/3 ).
It is easy to find an asymptotic distribution of V(τ ) by using our
heuristic normal approximation (1.16) in a way similar to (1.17)–(1.19).
2
(A) is almost constant, due to (1.26), we expect
Since the matrix σQ
2
that V(τ ) → N (0, σ0 τ I), where
8
(1.27)
σ02 =
.
3 Area(D)
BROWNIAN BROWNIAN MOTION
9
The following theorem shows that our heuristic estimate is correct:
Theorem 3. There is a function M0 = M0 (r) such that if r → 0 and
M → ∞, so that M > M0 (r), then we have weak convergence on the
interval 0 < τ < c:
(1.28)
V(τ ) → σ0 wD (τ )
and
(1.29)
Q(τ ) → Q0 + σ0
Z
τ
0
wD (s) ds
where wD (τ ) is a standard two dimensional Brownian motion subject
to the following modification: we set wD (τ ) = 0 for all τ > τ∗ , where
τ∗ is the earliest moment when the right hand side of (1.29) becomes
δ0 -close to ∂D.
1.4. Comparison to previous works. There are two directions of
research which our results belong to. The first one is the averaging
theory of differential equations, and the second one is the study of a
long time behavior of mechanical systems.
The averaging theory studies systems with two time scales, fast and
slow. The case where the fast motion is a Markov process, which does
not depend on the slow variables, is quite well understood [28]. The
results obtained in the Markov case has been extended to the situation
where the fast motion is given by a hyperbolic dynamical system in
[35, 44, 24]. By contrast, if the fast motion is coupled to the slow
one, as it is in our model, much less is known. Even the case where
the fast motion is a diffusion process was settled quite recently [42].
Some results are available for coupled systems where the fast motion is
uniformly hyperbolic [2, 36], but they all deal with short time intervals,
like the one in our Theorem 1. The behavior during longer time periods,
like those in our Theorems 2 and 3, is far less understood, and it might
be quite sensitive to the details of the problem at hand. For example,
the uniqueness of the limiting process in our Theorem 2 depends on the
2
smoothness of a certain auxiliary function, σQ
(A), cf. also Theorem 3
in [42]. More work is needed in this direction.
Let us now turn to the second research field. Probably, the simplest mechanical model where one can observe a non-trivial statistical
behavior is a periodic Lorenz gas. Bunimovich and Sinai [7] (see an
improved version in [9]) establish a Brownian motion approximation
for the position of a particle travelling in the periodic Lorenz gas with
finite horizon. Comparing our setting to that of [7, 9] we point out two
main differences: (i) we consider the dynamics during a relatively short
10
N. CHERNOV AND D. DOLGOPYAT
period of time, and (ii) in the context of Theorems 2–3, our system is
far from equilibrium.
On the other hand, models of Brownian motion where one massive
(tagged) particle is surrounded by an ideal gas of light particles are
studied in many papers, see, e.g., [12, 25, 26]. These models are more
realistic in explaining the actual Brownian Motion, however, there are
still some unresolved questions. For example, it is commonly assumed
that the ideal gas is (and remains) at equilibrium, but there is no satisfactory mathematical explanation of why and how the ideal gas reaches
and maintains the state of equilibrium. In ideal gases, where no direct
interaction between particles takes place, equilibrium can only establish
and propagate due to indirect interaction via collisions with the heavy
particle. This process requires many collisions of each gas particle with
the heavy particle, but the existing techniques are insufficient to analyze the dynamics beyond the time when each gas particle experiences
just a few collisions, cf. [19, 20]. Our model contains a single light
particle, but we are able to control the dynamics up to cM a collisions
(and actually longer, the main restriction of our analysis is the lack of
control over σQ (A) as the heavy disk approaches the border ∂D). We
hope that our method can be used to analyze many particle systems
as well.
We refer the reader to the surveys [29, 55, 54, 5] for description
of other models where macroscopic equations have been derived from
deterministic microscopic laws. Here we just mention that the main difficulty in deriving such equations is that microscopic equations of motion are reversible, while the limit macroscopic equations are not, and
therefore, the convergence is not possible everywhere in phase space.
Normally, the convergence occurs on a set of large (and, asymptotically,
full) measure, which is said to represent “typical” phase trajectories of
the system. In each problem, one has to carefully describe that large
subset of the phase space and estimate its measure. If the system has
some hyperbolic behavior, that large set has quite a complex fractal
structure.
The existing approaches to the problem of convergence make use of
certain families of measures on phase space, such that the convergence
holds with high probability with respect to each of those measures. In
the context of hyperbolic dynamical systems, the natural choice is the
family of measures having smooth conditionals on unstable manifolds
[52, 47, 45]. This works well for averaging problems when the hyperbolicity is uniform and the dynamics is smooth [35, 23, 24]. However,
if the stable and unstable manifolds have singularities, the situation is
much more complicated, and a detailed analysis of the dynamics near
BROWNIAN BROWNIAN MOTION
11
the singularities becomes necessary. Still, our paper shows that this
approach can work for systems with singularities. We hope that the
methods of our paper can be applied to other, more realistic, physical models, but here we restrict ourselves to the simplest non-trivial
model in order to illustrate our approach. We plan to address some
more realistic problems in future publications.
Acknowledgment. We would like to thank Ya. G. Sinai who suggested this problem to us, motivated by the studies of the dynamics of
a massive piston in an ideal gas [20]. Sinai outlined a general strategy
of the proof of Theorem 1, and we benefited from long conversations
with him about the entire work. N. Chernov was partially supported
by NSF grant DMS-0098788. D. Dolgopyat was partially supported by
NSF grant DMS-0245359. This work was done when the authors, in
turn, stayed the Institute for Advanced Study with partial support by
NSF grant DMS-9729992. We are grateful to the institute staff and
especially to our host T. Spencer for the excellent working conditions.
D. Dologopyat would like to thank D. Ruelle for explaining to him the
results of [48]. A local version of these results constitute an important
tool in this paper.
2. Plan of the proofs
2.1. General strategy. Our heuristic calculations of the asymptotic
distribution of V(τ ) in Section 1.3 were based on the normal approximation (1.16), hence we need to prove it. A natural approach is to
fix the heavy disk at Q = Q(t) and approximate the map F : Ω → Ω
by the billiard map FQ,V : ΩQ,V → ΩQ,V , which is known [7, 57] to
obey the central limit theorem (1.12). This approach, however, has
obvious limitations. First, we expect our map F to be hyperbolic, so
its nearby trajectories would diverge exponentially fast, hence this approximation would only work during time intervals O(ln M ), which are
far shorter than we need. The second limitation is more serious: the
central limit theorem (1.12) holds with respect to the billiard measure
µQ(t),V (t) , while we have to deal with the initial measure µQ0 ,V0 and its
images under our map F .
To overcome these limitations, we will show that the measures F n (µQ0 ,V0 )
can be well approximated (in the weak topology) by convex sums of
billiard measures µQ,V :
Z
n
(2.1)
F (µQ0 ,V0 ) ∼ µQ,V dλn (Q, V )
where λn is some factor measure.
12
N. CHERNOV AND D. DOLGOPYAT
Furthermore, it is convenient to work with a larger family of (auxiliary) measures, for which we can prove the convergence to equilibrium
in the sense of (2.1). We choose measures that have smooth conditional distributions on one dimensional curves in Ω, which are C 1 -close
to unstable manifolds of the billiard maps FQ,V . We call such curves
with appropriate densities on them standard pairs, see definition below. Then we extend the approximation (2.1) so that for each auxiliary
measure µ0
Z
n
0
(2.2)
F (µ ) ∼ µQ,V dλn (Q, V )
for large enough n, see Proposition 2.2 below.
The proof of the “equidistribution” (2.1)–(2.2) follows a shadowing
type argument developed in the theory of uniformly hyperbolic systems
without singularities [1, 23, 32, 48]. However, significant extra effort
is required to extend this argument to systems with singularities. In
fact, the largest error terms in our approximation (2.1) come from the
orbits passing near singularities.
There are two places where we have trouble establishing (2.1)–(2.2)
at all. First, if the velocity of the light particle becomes small, kvk ≈ 0,
then our system is no longer a small perturbation of a billiard dynamics
(because the heavy disk can move a significant distance between successive collisions with the light particle). Second, if the heavy disk comes
too close to the border ∂D, then the mixing properties of the corresponding billiard dynamics deteriorate dramatically (roughly speaking,
because the light particle can be trapped in a narrow tunnel between
∂P and ∂D for a long time), and the central limit theorem only holds
at very large times, during which the position and velocity of the heavy
disk can change significantly. We will show that the first phenomenon
is improbable on the time scale we deal with. However, the second
phenomenon forces us to stop the heavy disk when it comes too close
to the border ∂D.
Our paper can be divided, roughly, into two parts. Section 3 and
Appendices A and B are mostly “dynamical”, they involve specifics of
our mechanical model of two particles in order to construct auxiliary
measures and prove the equidistribution (2.1)–(2.2). On the contrary,
Sections 4–6 are mostly probabilistic, they consist of the derivation of
asymptotical results claimed in Theorems 1, 2 and 3 by using standard
moment-type techniques1 [31, 22].
1Note
that the time scale in√
[22] corresponds to that of our Theorem 2. Indeed,
in the notation of [22], ε = 1/ M is the typical velocity of the heavy particle at
equilibrium, hence their “4/3 law” becomes our “2/3 law”. Let us also mention the
BROWNIAN BROWNIAN MOTION
13
We emphasize that the specifics of the underlying model are only
used in the construction of auxiliary measures and in the proof of
Proposition 2.2. Hence, if one establishes similar results for another
system, one would be able to derive analogues of our limit theorems in
that setting.
2.2. Main steps in the proofs Theorem 2. The proofs of Theorems 1–3 follow similar lines, but Theorem 2 requires more effort than
the other two (in particular, because we are unable to obtain an explicit formula for the limiting process and so we have to proceed in
a roundabout way). We divide the proof of Theorem 2 between two
sections: the convergence to equilibrium in the sense of (2.1)–(2.2) is
established in Section 3, and the the moment estimates specific to the
scaling of Theorem 2 are done in Section 4. The modifications needed
to prove Theorems 1 and 3 are described in Sections 5 and 6.
Next we specify the main steps in the proof of (2.1)–(2.2) beginning
with the definition of a family of auxiliary measures.
Recall that our primary goal is control over measures F n (µQ,V ) for
n ≥ 1. The measure µQ,V is concentrated on the surface ΩQ,V . Let
us coarse-grain this measure by partitioning ΩQ,V into small subdomains D ⊂ ΩQ,V and representing µQ,V as a sum of its restrictions to
those domains. The image of a small domain D ⊂ ΩQ,V under the
map F n gets strongly expanded in the unstable direction of the billiard map FQ,V , strongly compressed in the stable direction of FQ,V ,
slightly deformed in the transversal directions, and possibly cut by singularities of F into several pieces. Thus, F n (D) looks like a union of
one-dimensional curves that resemble unstable manifolds of the billiard
map FQ,V . Thus, the measure F n (µQ,V ) evolves as a weighted sum of
smooth measures on such curves.
Motivated by this observation we introduce our family of auxiliary
measures. A standard pair is ` = (γ, ρ), where γ ⊂ Ω is a C 2 curve,
which is C 1 close to an unstable manifold of the billiard map FQ,V for
some Q, V , and ρ is a probability density on γ. The precise description
of standard pairs is given in Section 3, here we only mention the properties of standard pairs most essential to our analysis. For a standard
pair `, we denote by γ` its curve, by ρ` its density, and by mes` the
corresponding measure on γ` .
papers [33, 34] studying the exit problem from a neighborhood of a non-degenerate
equilibrium perturbed by a small noise. Our Theorem 2 deals with a degenerate
equilibrium, but the heuristic argument used to determine the correct scaling is
similar to [33, 34].
14
N. CHERNOV AND D. DOLGOPYAT
We define auxiliary measures via convex sums of measures on standard pairs, which satisfy an additional “length control”:
Definition. An auxiliary measure m on Ω is such that
(2.3)
m = m 1 + m2 ,
|m2 | < CM −50
(here C > 0 is a large fixed constant) and m1 is given by
Z
(2.4)
m1 = mes`α dλ(α)
where {`α = (γα , ρα )} is a family of standard pairs such that {γα } make
a measurable partition of Ω (m1 -mod 0), and λ is some factor measure
satisfying
(2.5)
λ α : length(γα ) < M −100 = 0
which imposes a length control. We denote by M the family of auxiliary
measures.
It is clear that our family M contains the initial smooth measure
µQ√
0 ,V0 , as well as every billiard measures µQ,V for Q, V satisfying kV k <
1/ M and dist(Q, ∂D) > r + δ0 /2. Indeed, one can easily represent
any of these measures by its conditional distributions on the fibers of
a rather arbitrary smooth foliation of the corresponding space ΩQ,V
into curves whose tangent vectors lie in unstable cones, see precise
definitions in Section 3.
Next we need to define a class of functions R = {A : Ω → R}
satisfying two general (and somewhat conflicting) requirements. On
the one hand, the functions A ∈ R should be smooth enough on the
bulk of the space Ω to ensure a fast (in our case – exponential) decay of
correlations under the maps FQ,V . On the other hand, the regularity of
the functions A ∈ R should be compatible with that of the map F , so
that for any A ∈ R the function A ◦ F would have regularity properties
similar to those of A.
Our map F : Ω → Ω is not smooth, its singularity set S = F −1 (∂Ω)
consists of points whose next collision is grazing. Due to our “finite
horizon” assumption, S ⊂ Ω is a finite union of compact C 2 smooth
submanifolds (with boundaries). We note that while the map F depends on the mass M of the heavy disk, the singularity set S does not.
The complement Ω \ S is a finite union of open connected domains, we
call them Ωk , 1 ≤ k ≤ k0 . The restriction of the map F to each Ωk is
C 2 . The derivatives of F are unbounded, but their growth satisfies the
following inequality:
(2.6)
kDx F k ≤ LF · [dist(x, S)]−1/2
BROWNIAN BROWNIAN MOTION
15
where LF > 0 is independent of M , see a proof Section 3.1. In addition,
the restriction of F to each Ωk can be extended by continuity to the
closure Ω̄k , it then loses smoothness but remains Hölder continuous:
(2.7)
∀k
∀x, y ∈ Ω̄k
kF (x) − F (y)k ≤ KF [dist(x, y)]1/2
where KF > 0 is independent of M , see a proof Section 3.1.
These facts lead us to the following definition of R:
Definition. A function A : Ω → R belongs to R iff
(a) A is continuous everywhere except, possibly, the set S. Moreover,
the continuous extension of A to the closure of each connected component Ωk of Ω \ S is Hölder continuous with some exponent αA ∈ (0, 1]:
∀k
∀x, y ∈ Ω̄k
|A(x) − A(y)| ≤ KA [dist(x, y)]αA
(b) at each point x ∈ Ω\S the function A has a local Lipschitz constant
(2.8)
Lipx (A) := lim sup |A(y) − A(x)|/dist(x, y)
y→x
which satisfies the restriction
Lipx (A) ≤ LA [ dist(x, Sn )]−βA
with some βA < 1. The quantities αA ≤ 1, βA < 1, and KA , LA > 0
may depend on the function A.
Note that the set S, and hence the class R are independent of M .
In Appendix A we will prove the following:
Lemma 2.1. Let n ≥ 1 and B1 , B2 ∈ R. Then the function A =
B1 (B2 ◦ F n−1 ) has the following properties:
(a) A is continuous everywhere except for the set Sn on which the map
F n itself is discontinuous. Moreover, the continuous extension of A to
the closure of each connected component Ωk,n of the complement Ω \ Sn
is Hölder continuous with some exponent αA ∈ (0, 1]:
∀x, y ∈ Ω̄k,n
|A(x) − A(y)| ≤ KA [dist(x, y)]αA
(b) at each point x ∈ Ω \ Sn the local Lipschitz constant (2.8) of A
satisfies the restriction
Lipx (A) ≤ LA [ dist(x, Sn )]−βA
with some βA < 1. Here αA , βA , KA , and LA are determined by n and
αBi , βBi , KBi , LBi for i = 1, 2, but they do not depend on M .
For any function A : Ω → R and a standard pair ` = (γ, ρ) we shall
write
Z
E` (A) = A(x)ρ(x) dx.
γ
16
N. CHERNOV AND D. DOLGOPYAT
We also denote π1 (Q, V, q, v) = (Q, V ). In the following proposition,
0 < δ1 < δ0 and c δ1 are small constants, and K > 0 is a large
constant.
Proposition 2.2. Let ` = (γ, ρ) be a standard pair such that
M V̄ 2 ≤ 1 − δ1
(2.9)
D(Q̄, ∂D) > r + δ1
for some (Q̄, V̄ ) ∈ π1 (γ). Let A = B1 (B2 ◦F n0 −1 ) for some B1 , B2 ∈ R
and n0 ≥ 1. Then
√
(a) (Invariance). If n ∈ [K | ln length(γ)|, c M ], then
X
E` (A ◦ F n ) =
cα E`α (A)
α
P
where cα > 0, α cα = 1, `α = (γα , ρα ) are standard pairs (the components of the image of ` under F n )
X
√
cα ≤ C ε
length(γα )<ε
for all ε > 0, the map F −n is smooth on each γα , and
(2.10)
∀y 0 , y 00 ∈ γα
dist[F −m (y 0 ), F −m (y 00 )] ≤ Kθ m
for all 1 ≤ m ≤ n and some constant θ < 1.
(b) (Short term equidistribution). If n ∈ [K | ln length(γ)|, M a ] for
some small a > 0 (precisely, for any a < 1/4), then ∀m ≤ min{n/2, K ln M }.
E` (A ◦ F n ) = µQ̄,V̄ (A) + O(Rn,m + θ m ),
where
Rn,m = kV̄ k(n + m2 ) + (n2 + m3 )/M
h
√ i
(c) (Long term equidistribution). Let n ∈ K | ln length(γ)|, c M .
Assume that Ā = µQ,V (A) is independent of Q, V . Then for all j ∈
[K | ln length(γ)|, M a ] and m ≤ min{j/2, K ln M } we have
E` (A ◦ F n ) = Ā + O(Rn,j,m + θ m ),
where
Rn,j,m = E` (kVn−j k) (j + m2 ) + (j 2 + m3 )/M,
and Vn−j denotes the V component of the point F n−j (x), x ∈ γ.
BROWNIAN BROWNIAN MOTION
17
Observe that the equidistribution property, combined with the fact
µQ,V ∈ M, implies (2.1).
Theorem 2 follows from the next proposition that we formulate in
probabilistic terms (however, it is known to be equivalent to the fact
that limiting factor measure λn (Q, V ) of (2.1) satisfies an associated
partial differential equation).
Proposition 2.3. Let M0 > 0 and a > 0. The families of random
processes Q∗ (τ M 2/3 ) and M 2/3 V∗ (τ M 2/3 ) such that M ≥ M0 , and the
initial condition (Q0 , V0 , q(0), v(0)) is chosen randomly with respect to
a measure in M such that almost surely kV0 k ≤ aM −2/3 , are tight and
any limit process (Q(τ ), V(τ )) satisfies (1.24).
The proofs of Propositions 2.2 and 2.3 are given in Sections 3 and 4,
respectively. In Sections 5 and 6 we describe the modifications needed
to prove Theorems 1 and 3 respectively. Section 5 is especially short
since the material there is quite similar to [24], Sections 13 and 14,
except that here some additional complications are due to the fact
that we have to deal with continuous time system.
3. Standard pairs and equidistribution
The main goals of this section are the construction of standard pairs
the proof of Proposition 2.2.
3.1. Unstable vectors. Proposition 2.2 requires that
(3.1)
(Q, V ) ∈ Υδ1 := {dist(Q, ∂D) > r + δ1 , M kV k2 < 1 − δ1 }
and we restrict our analysis to this region. We first discuss the flow Φt
on the full (seven-dimensional) phase space M in order to collect some
preliminary estimates.
Let x = (Q, V, q, v) ∈ M be an arbitrary point and dx = (dQ, dV, dq, dv) ∈
Tx M a tangent vector. Let dx(t) = DΦt (dx) be the image of dx at
time t. We describe the evolution of dx(t) for t > 0.
Between successive collisions, the velocity components dV and dv
remain unchanged, while the position components are updated linearly:
(3.2)
dQ(t + s) = dQ(t) + s dV (t),
dq(t + s) = dq(t) + s dv(t)
At collisions, the tangent vector dx(t) changes discontinuously, as we
describe below.
First, we need to introduce convenient notation. For any unit vector
n ∈ R2 (usually, a normal vector to some curve), we denote by Gn the
projection onto n, i.e. Gn (u) = hu, ni n, and by Hn the projection onto
18
N. CHERNOV AND D. DOLGOPYAT
the line perpendicular to n, i.e. Hn (u) = u − Gn (u). Also, denote by
Rn the reflection across the line perpendicular to n, that is
Rn (u) = −Gn (u) + Hn (u) = u − 2hu, ni n
For any vector w 6= 0, the operator Hw/kwk we write as Hw , for brevity.
Now, consider a collision of the light particle with the border ∂D,
and let n denote the inward unit normal vector to ∂D at the point of
collision. The components dQ and dV remain unchanged because the
heavy disk is not involved in this event. The basic rule of specular
reflection at ∂D reads v + = Rn (v − ) (the superscripts “+” and “−” refer to the postcollisional and precollisional vectors, respectively). Note
that kv + k = kv − k. Accordingly, the tangent vectors dq and dv change
by
dq + = Rn (dq − )
and
where
dv + = Rn dv − + Θ+ (dq + )
2κ kv + k2
Hv +
hv + , ni
Here κ denotes the (positive) curvature of the boundary at the point of
reflection. Note that kdq + k = kdq − k and Θ+ (dq + ) = Θ− (dq − ) where
Θ+ =
(3.3)
Θ− =
2κ kv + k2
Rn ◦ H v −
hv + , ni
Also, the geometry of reflection implies hv + , ni > 0.
Next, consider a collision between the two particles. At the moment
of collision we have q ∈ ∂P(Q), i.e. kq − Qk = r. Let n = (q − Q)/r
be the normalized relative position vector. Then the laws of elastic
collision (1.1)–(1.2) can be written as
2M
Gn (v − − V − )
M +1
2M
1
−
−
−
= Rn (v ) +
Gn (v ) + Gn (V )
M +1 M
v+ = v− −
and
2
Gn (v − − V − )
M +1
Let w = v − V denote the relative velocity vector. We note that
V+ = V−+
w + = w − − 2 Gn (w − ) = Rn (w − )
BROWNIAN BROWNIAN MOTION
19
and hence kw + k = kw − k. The components dq and dQ of the tangent
vector dx change according to
2M
1
+
−
−
−
dq = Rn (dq ) +
Gn (dq ) + Gn (dQ )
M +1 M
dQ
+
1
2M
−
−
Gn (dq ) + Gn (dQ )
= Rn (dQ ) +
M +1 M
2
= dQ− +
Gn (dq − − dQ− )
M +1
−
Note that
dq + − dQ+ = Rn (dq − − dQ− )
and so kdq + − dQ+ k = kdq − − dQ− k. Next, the components dv and
dV of the tangent vector dx change by
2M
1
+
−
−
−
dv = Rn (dv ) +
Gn (dv ) + Gn (dV )
M +1 M
M
Θ+ (dq + − dQ+ )
+
M +1
dV + = dV − +
−
2
Gn (dv − − dV − )
M +1
1
Θ+ (dq + − dQ+ )
M +1
where
2κ kw + k2
Hw +
hw + , ni
Here κ = 1/r is the curvature of ∂P (Q). Note that Θ+ (dq + − dQ+ ) =
Θ− (dq − − dQ− ), where
Θ+ =
Θ− =
2κ kw +k2
Rn ◦ H w −
hw + , ni
Also, the geometry of collision implies hw + , ni > 0, since we have
chosen n to point toward the light particle.
All the above equations can be verified directly. Alternatively, we
can use the fact that our system of two particles of different masses
M 6= m reduces to a billiard
the
√ dimensional
√ in a four
√ domain by √
change of variables Q̃ = Q M , Ṽ = V M , q̃ = q m, and ṽ = v m
(the latter two are trivial since m = 1). This reduction is standard
[51], and we omit details. Then the above equations can be derived
from the general theory of billiards [14, 37, 51].
20
N. CHERNOV AND D. DOLGOPYAT
Now, since the total kinetic energy is fixed (1.3), the velocity components dv and dV of the tangent vector dx satisfy
(3.4)
hv, dvi + M hV, dV i = 0
In addition, the Hamiltonian character of the dynamics implies that if
the identity
(3.5)
hv, dqi + M hV, dQi = 0
holds at some time, it will be preserved at all times (future and past).
From now on, we assume that all our tangent vectors satisfy (3.5).
There is a class of tangent vectors, which we will call unstable vectors,
that is invariant under the dynamics. It is described in the following
proposition:
Proposition 3.1. The class of tangent vectors dx with the following
properties remains invariant under the forward dynamics:
(a) hdq, dvi ≥ (1 − C −1 ) kdqk kdvk
C
(b) kdQk ≤ M
kdqk
C
(c) kdV k ≤ M kdvk
(d) hdq, vi ≤ CkV k kdqk ≤ √CM kdqk
(e) hdv, vi ≤ CkV k kdvk ≤ √CM kdvk
(f) kdqk ≤ Ckdvk
(g) kdvk ≤ |t−C(x)| kdqk
(h) Equations (3.4) and (3.5) hold.
Here C > 1 is a large constant, and t− (x) = max{t ≤ 0 : Φt (x) ∈ Ω}
is the time of the latest collision along the past trajectory of x.
The proof of this proposition is based on the previous equations and
some routine calculations, which we omit.
Unstable vectors have strong (uniform in time) expansion property:
Proposition 3.2. Let dx be an unstable tangent vector and dx(t) =
DΦt (dx) its image at time t > 0. Then the norm kdx(t)k monotonically
grows with t. Furthermore, there is a constant θ < 1 such that for any
two successive moments of collisions t < t0 of the light particle with
∂D ∪ ∂P(Q) we have
(3.6)
kdx(t + 0)k ≤ θkdx(t0 + 0)k
The notation t+0, t0 +0 mean that the postcollisional vectors are taken.
The proof easily follows from the previous equations. In fact,
(3.7)
θ −1 = 1 + Lmin κmin
BROWNIAN BROWNIAN MOTION
21
where κmin > 0 is the smaller of 1/r and the minimal curvature of ∂D,
Lmin is the smaller of the minimal distance between the scatterers and
δ1 , the minimal distance from the heavy disk to the scatterers allowed
by (3.1).
When the light particle collides with ∂D and ∂P(Q), it is more
convenient to use the vector w defined by (1.4), instead of v, and respectively dw instead of dv. Then the vector w will changes by the
same rule w + = Rn (w − ) for both types of collisions. At collisions with
∂D, the vector dw = dv will change by the rule
dw + = Rn dw − + Θ+ (dq + )
while at collisions with the heavy disk, the vector dw = dv − dV will
change by a similar rule
dw + = Rn dw − + Θ+ (dq + − dQ+ )
Furthermore, the expressions for Θ+ and Θ− will be identical for both
types of collisions.
It is easy to see that the inequalities (a)–(g) in Proposition 3.1 remain
valid if we replace v by v − V and dv by dv − dV at any phase point,
hence they apply to the vectors w and dw at the points of collision.
The inequality (3.6) will also hold in the norm on Ω defined by
(3.8)
kdxk2 = kdQk2 + kdV k2 + kdqk2 + kdwk2
Remark. Our equations show that the postcollisional tangent vector
(dQ+ , dV + , dq + , dw + ) depends on the precollisional vector (dQ− , dV − , dq − , dw − )
smoothly, unless hw + , ni = 0. This is the only cause of singularity, it
corresponds to grazing collisions (also, colloquially, called “tangential
collisions”).
Remark. Our equations imply that the derivative of the map F is
bounded by kDx F k ≤ Const/hw + , ni, where Const does not depend
on M . It is easy to see that dist(x, S) = O(hw + , ni2 ), hence we obtain
(2.6). Now (2.7) easily follows by integrating (2.6).
3.2. Unstable curves. We call a smooth curve W ⊂ M an unstable
curve (or u-curve, for brevity) if, at every point x ∈ W, the tangent vector to W is an unstable vector. By Propositions 3.1 and 3.2 the future
image of a u-curve is a u-curve, which may be only piecewise smooth,
due to singularities, and that u-curve is expanded by Φt monotonically
and exponentially fast in time.
Now we extend our analysis to the collision map F : Ω → Ω defined
in Section 1.2. For every point x ∈ M we denote by t+ (x) = min{t ≥
0 : Φt (x) ∈ Ω} and t− (x) = max{t ≤ 0 : Φt (x) ∈ Ω} the first
22
N. CHERNOV AND D. DOLGOPYAT
collision times in the future and the past, respectively. Let Ψ± (x) =
±
Φt (x) (x) ∈ Ω denote the respective “first collision” maps. Note that
F (x) = Ψ+ (Φε x) for all x ∈ Ω and small ε > 0.
For any unstable curve W ⊂ M, the projection W = Ψ− (W) is a
smooth or piecewise smooth curve in Ω, whose components we also call
u-curves. Let dx = (dQ, dV, dq, dw) be the postcollisional tangent vector to W at a moment of collision. Its projection under the derivative
DΨ− is a tangent vector dx0 = (dQ0 , dV 0 , dq 0 , dw 0 ) to the curve W ⊂ Ω.
Observe that dV 0 = dV , dw 0 = dw, dQ0 = dQ − tV , and dq 0 = dq − tv,
where t is uniquely determined by the condition dx0 ∈ Tx Ω. Some
elementary geometry and application of Proposition 3.1 give
Proposition
3.3. There is a constant 1 < C < ∞ such that kdQ0 k ≤
√
Ckdvk/ M and C −1 kdvk ≤ kdq 0 k ≤ Ckdvk. Therefore,
C −1 ≤
kdx0 k
≤C
kdxk
We introduce two norms (metrics) on u-curves W ⊂ Ω. First, we
denote by length(·) the norm on W induced by the Euclidean norm
kdx0 k on Tx (Ω). Second, if W = Ψ− (W), we denote by | · | the norm
on W induced by the norm (3.8) on the postcollisional tangent vectors
dx to W at the moment of collision. Due to the last proposition, these
norms are equivalent in the sense
(3.9)
C −1 ≤
length(W )
≤C
|W |
By (3.9), we can replace length(γ) with |γ| in the assumptions of
Proposition 2.2, as well as in many other estimates of our paper. We
actually prefer to work with the |·|-metric, because it has an important
uniform expansion property: the map F expands every u-curve in the
| · |-metric by a factor ≥ θ −1 > 1, see (3.7) (note that the length(·)
metric lacks this property).
Observe that the Q, V coordinates vary along u-curves W ⊂ Ω very
slowly, so that u-curves are almost parallel to the cross-sections ΩQ,V
of Ω defined by (1.7). Each ΩQ,V can be supplied with standard coordinates. Let r be the arc length parameter along ∂D ∪ ∂P(Q) and
ϕ ∈ [−π/2, π/2] the angle between the vector w̄ and the normal vector
n. The orientation of r and ϕ is shown in Fig. 1. Topologically, ΩQ,V
is a union of cylinders, in which the cyclic coordinate r runs over the
boundaries of the scatterers and ∂P(Q), and ϕ ∈ [−π/2, π/2]. We
need to fix reference points on each scatterer and on ∂P(Q) in order to
BROWNIAN BROWNIAN MOTION
23
define r, and then the coordinate chart r, ϕ in ΩQ will actually be the
same for all Q, V . We denote by Ω0 that unique r, ϕ coordinate chart.
f
Figure 1: A collision of the light particle with a scatterer: the
orientation of r and ϕ.
Note that r and ϕ are defined at every point x ∈ Ω, hence they
make two coordinates in the (six-dimensional) space Ω. Since cos ϕ =
hw, ni/kwk, the singularities of the map F correspond to cos ϕ = 0,
i.e. to ϕ = ±π/2 (which is the boundary of Ω0 ). Let π0 denote the
natural projection of Ω onto Ω0 . Note that, for each Q, V the projection
π0 : ΩQ,V → Ω0 is one-to-one. Then the map
−1
πQ,V := π0 |ΩQ,V
◦ π0
defines a natural projection Ω → ΩQ,V (geometrically, it amounts to
moving the center of the heavy disk to Q, setting its velocity to V , and
rescaling the vector w at points q ∈ ∂P(Q) by the rule (1.6)).
We turn back to u-curves W ⊂ M. For any such curve, W =
π0 (Ψ− (W)) is a smooth or piecewise smooth curve in Ω0 , whose components we also call u-curves. Any such curve is described by a smooth
function ϕ = ϕ(r). Let dx = (dQ, dV, dq, dv) be the postcollisional tangent vector to W at a moment of collision. Its projection under the
derivative D(π0 ◦ Ψ− ) is a tangent vector to W , which we denote by
(dr, dϕ).
To evaluate (dr, dϕ), we introduce two useful quantities, ∆ and B,
at each collision point. Let ∆ = kHw (dq)k if the light particle collides
with ∂D, and ∆ = kHw (dq − dQ)k if it collides with the disk. Then let
B=
kHw (dw)k
∆ kwk
24
N. CHERNOV AND D. DOLGOPYAT
Proposition 3.4. In the above notation, |dr| = cos∆ϕ and dϕ/dr =
B cos ϕ − κ, where κ > 0 is the curvature of ∂D ∪ ∂P at the point of
collision. There is a constant 1 < C < ∞ such that for any u-curve
W ⊂ Ω and any point x ∈ W
2κ
2κ
≤B≤
+C
cos ϕ
cos ϕ
and
C −1 ≤
(3.10)
dϕ
≤C
dr
In particular, dϕ/dr > 0, hence π0 (W ) is an increasing curve in the
r, ϕ coordinates. Lastly,
C −1 ≤
(dr)2 + (dϕ)2
≤C
kdxk2
The proof is based on elementary geometric analysis, and we omit
it.
Next we study the evolution of u-curves under the map F . Let
W0 ⊂ Ω be a u-curve on which F n is smooth for some n ≥ 1. Then
Wi = F i (W0 ) for i ≤ n are u-curves. Pick a point x0 ∈ W0 and
put xi = F i (x0 ) for i ≤ n. For each i, we denote by ri , ϕi , κi , Bi ,
etc. the corresponding quantities, as introduced above, at the point
xi . For i = 0, . . . , n − 1, denote by J(xi ) the Jacobian of the map
F : Wi → Wi+1 at the point xi in the norm | · |, i.e. the local expansion
factor of the curve Wi under F in the | · |-metric.
Proposition 3.5. There is a constant 1 < C < ∞ such that in the
above notation
1+
C
C −1
< J(xi ) < 1 +
cos ϕi+1
cos ϕi+1
and
(3.11)
J(x0 ) · · · J(xn−1 ) ≥ θ −n
where θ < 1 is given by (3.7).
Remark. By setting M = ∞ in all our results we obtain their analogues
for the billiard-type dynamics in D \ P(Q) in which the disk P(Q) is
fixed and the light particle moves at a constant speed kw̄k = SV given
by (1.6). Thus we recover standard definitions of unstable vectors
BROWNIAN BROWNIAN MOTION
25
and unstable curves for the billiard-type map FQ,V : ΩQ,V → ΩQ,V .
Proposition 3.4 implies that the kdxk norm on ΩQ,V becomes
(3.12)
kdxk2 = kdqk2 + kdwk2
= (dr cos ϕ)2 + SV2 (dϕ + κ dr)2
In this norm, the map FQ,V expands every unstable curve by a factor
≥ θ −1 > 1.
As usual, reversing the time gives the definition of stable vectors and
stable curves (or s-curves for brevity). Those are decreasing in the r, ϕ
coordinates and satisfy the bound
(3.13)
−C < dϕ/dr < −C −1 < 0
The corresponding norm on stable vectors/curves must be defined on
precollisional tangent vectors and is expressed by
(3.14)
kdxk2stable = (dr cos ϕ)2 + SV2 (dϕ − κ dr)2
In this norm, the map FQ,V contracts every s-curve by a factor ≤ θ < 1.
3.3. Homogeneous unstable curves. To control distortions of ucurves by the map F , we need to carefully partition the neighborhood of
the singularity set cos ϕ = 0 into countably many surrounding sections
(shells). This procedure has been introduced in [9] and goes as follows.
Fix a large k0 ≥ 1 and for each k ≥ k0 define two “homogeneity strips”
in Ω0
Ik = {(r, ϕ) : π/2 − k −2 < ϕ < π/2 − (k + 1)−2 }
and
I−k = {(r, ϕ) : −π/2 + (k + 1)−2 < ϕ < −π/2 + k −2 }
We also put
I0 = {(r, ϕ) : −π/2 + k0−2 < ϕ < π/2 − k0−2 }
Slightly abusing notation, we will also denote by I±k the sets π0−1 (I±k ) ⊂
Ω and call them homogeneity sections. A u-curve W ⊂ Ω is said to be
weakly homogeneous if W belongs to one section Ik for some |k| ≥ k0
or for k = 0.
Now let W0 ⊂ Ω be a u-curve on which F n is smooth, and assume
that the u-curve Wi = F i (W0 ) is weakly homogeneous for every i =
0, 1, . . . , n. Let x0 , x00 ∈ W0 be two points and put xi = F i (x0 ) and
x0i = F i (x00 ) for 1 ≤ i ≤ n. We denote by ri , ϕi , κi , Bi , etc. the
corresponding quantities, as introduced above, at the point xi , and
similarly ri0 , ϕ0i , κ0i , Bi0 , etc. at the point x0i . For i = 0, . . . , n − 1, denote
by J(xi ) and J(x0i ) the Jacobians of the map F : Wi → Wi+1 at the
points xi and x0i , respectively, in the | · |-norm. Let Wi (xi , x0i ) denote
the segment of the curve Wi between the points xi and x0i . Lastly, let
26
N. CHERNOV AND D. DOLGOPYAT
∠(xi , x0i )W be the angle between the tangent vectors to the curve Wi
at the points xi and x0i (in the manifold Ω).
Proposition 3.6 (Distortion bounds). Under the above assumptions,
if the following bound holds for i = 0 with some C0 = c > 0, then
it holds for all i = 1, . . . , n − 1 with some Ci = C > c (i.e., Ci is
independent of i and n)
0
J(xi ) ln
≤ Ci |Wi+1 (xi+1 , xi+1 )|
J(x0 ) |Wi+1 |2/3
i
Moreover, in this case
0
J(x0 ) · · · J(xn−1 ) ln
≤ C |Wn (xn , xn )|
J(x0 ) · · · J(x0 ) |Wn |2/3
0
n−1
Proposition 3.7 (Curvature bounds). Under the above assumptions,
if the following bound holds for i = 0 with some C0 = c > 0, then it
holds for all i = 1, . . . , n with some Ci = C > c (independent of i and
n)
|Wi (xi , x0i )|
∠(xi , x0i )W ≤ Ci
|Wi |2/3
The proofs of these two propositions are quite lengthy. They are
given in Appendix B.
Consider now the case where either F n is not smooth on the original
u-curve W0 or the u-curve Wi is not weakly homogeneous (i.e. it crosses
the boundaries of homogeneity sections) for some i = 0, . . . , n. Then
we need to partition W0 into a finite or countable collection of u-curves
W0,n,j such that Wi,n,j = F i (W0,n,j ) is a weakly homogeneous u-curve
for each i = 0, . . . , n and every j. We use the unique maximal partition
defined by this requirement (it can be constructed by induction on n,
assuming that the boundaries of the homogeneity sections act as additional singularities). For every n ≥ 0, we call Wn,n,j the homogeneous
components (or H-components, for brevity) of the image F n (W0 ).
Observe that if the curve W0 satisfies the distortion bound and the
curvature bound for i = 0, then so does any part of it (because |W0 |
and |W1 | decrease if we reduce the size of the curve). Therefore, if a
weakly homogeneous u-curve satisfies the distortion bound and curvature bound for i = 0, then those bounds apply to all the H-components
of its future images for all 0 ≤ i ≤ n < ∞, as it follows from Propositions 3.6 and 3.7. Thus we can restrict our studies to weakly homogeneous u-curves that satisfy the distortion and curvature bounds:
BROWNIAN BROWNIAN MOTION
27
Definition. A weakly homogeneous u-curve W0 is said to be homogeneous (or an H-curve, for brevity) if it satisfies the above distortion
bound and curvature bound for i = 0 (hence its H-components will
satisfy these bounds as well).
We note that |Wn (xn , x0n )|/|Wn |2/3 ≤ |Wn |1/3 ≤ Const, hence the
distortions of H-curves under the maps F n , n ≥ 1, are uniformly
bounded, in particular, for some constant β̃ > 1
(3.15)
e−β̃
0
|W0 (x0 , x00 )|
|Wn (xn , x0n )|
β̃ |W0 (x0 , x0 )|
≤
≤
e
|F −n (Wn )|
|Wn |
|F −n (Wn )|
Now consider an H-curve W0 such that Wi = F i (W0 ) is an H-curve
for every i = 1, . . . , n. Let mes0 be an absolutely continuous measure
on W0 with density ρ0 (with respect to the measure induced by the
| · |-norm). Then mesi = F i (mes0 ) is a measure on the curve Wi with
some density ρi for each i = 1, . . . , n. As an immediate consequence of
Proposition 3.6 and (3.15), we have
Corollary 3.8 (Density bounds). Under the above assumptions, if the
following bound holds for i = 0 with some C0 = c > 0, it holds for all
i = 1, . . . , n with some Ci = C > c (independent of i and n)
0
ρi (xi ) ln
≤ Ci |Wi (xi , xi )|
(3.16)
ρi (x0 ) |Wi |2/3
i
Observe that if the density bound holds for i = 0 on the curve W0 ,
then it holds on any part of it (because |W0 | decreases as we reduce
the size of the curve). Therefore, if F n is not smooth on the original
H-curve W0 or if Wi is not an H-curve for some i = 1, . . . , n, then the
density of the restriction of the measure mesi to every H-component
of F i (W0 ) will satisfy the above estimate. Hence we can restrict our
studies to densities satisfying this estimate:
Definition. Given an H-curve W0 , we say that ρ0 is a homogeneous
density if it satisfies the bound (3.16) with i = 0 (hence the induced
densities of the measure F n (mes0 ) on the H-components of F n (W0 )
will satisfy this same bound for all n ≥ 1).
Note that the above estimate remains valid whether we normalize our
measures or not. Also observe that for any homogeneous density ρ(x)
on any H-curve W , the function ln ρ(x) is Lipschitz continuous, but its
Lipschitz constant is not uniform, it may be arbitrarily large, because
W may be arbitrarily short. However, because |W (x, x0 )|/|W |2/3 <
28
N. CHERNOV AND D. DOLGOPYAT
|W |1/3 < Const, we have a uniform bound
(3.17)
e−β̃ ≤
ρ(x)
≤ eβ̃ ,
ρ(x0 )
∀x, x0 ∈ W
where β̃ = C0 maxW |W |1/3 . Moreover, Proposition 3.7 implies
(3.18)
∠(x, x0 )W ≤ β̃,
∀x, x0 ∈ W
It will be convenient to impose a tight upper bound on the length of
H-curves requiring that they must be shorter than a small δ̃ > 0 (larger
H-curves can be partitioned into H-curves of length between δ̃/2 and
δ̃ > 0, in an arbitrary manner). We fix a small δ̃ > 0 to make β̃ in
(3.15), (3.17) and (3.18) small enough. This will make our H-curves
almost straight lines, the map F on them will be almost linear, and
the homogeneous densities will be almost constant.
3.4. Standard pairs. Now we define standard pairs mentioned earlier
in Section 2.2:
Definition. A standard pair ` = (γ, ρ) is an H-curve γ ⊂ Ω with a
homogeneous probability density ρ on it. We denote by mes = mes ` the
measure on γ with density ρ.
Our previous results imply the following invariance of the class of
standard pairs:
Proposition 3.9. Let ` = (γ, ρ) be a standard pair. Then for each n ≥
0, every smooth curve γ 0 ⊂ F n (γ) with the density ρ0 of the measure
F n (mes` ) conditioned on γ 0 is a standard pair.
Recall that F expands H-curves by a factor ≥ θ −1 > 1, which is
a local property. It is also important to show that H-curves grow
in a global sense, i.e. given a small H-curve γ, the sizes of the Hcomponents of F n (γ) tend to grow exponentially in time until they
become of order one, on the average (we will make this statement
precise). Such statements are usually referred to as “growth lemmas”
[9, 57, 16, 17].
Let (γ, ρ) be a standard pair and mes denote the probability measure
with density ρ. For n ≥ 1 and x ∈ γ, let rn (x) be the distance from the
point F n (x) to the nearest endpoint of the H-component γn (x) ⊂ F n (γ)
that contains F n (x).
Lemma 3.10 (“Growth lemma”). Under the above conditions,
(a) There are constants β1 ∈ (0, 1) and β2 > 0, such that for any ε > 0
mes(x : rn (x) < ε) ≤ (β1 /θ)n mes(x : r0 < εθ n ) + β2 ε
BROWNIAN BROWNIAN MOTION
29
(b) There are constants β3 , β4 > 0, such that if n ≥ β3 ln |γ|, then for
any ε > 0 we have mes(x : rn (x) < ε) ≤ β4 ε.
(c) There are constants β5, β6 > 0, a small ε0 > 0, and q ∈ (0, 1) such
that for any n2 > n1 > β5 ln |γ| we have
mes x : max ri (x) < ε0 ≤ β6 q n2 −n1
n1 <i<n2
Proof. First, we prove (a) along the lines of Theorem 3.1 in [16]. There
are two competing factors that affect the evolution of the function rn (x)
on γ, as n grows. On the one hand, the expansion of H-curves under
F tends to increase rn (x) by a factor ≥ θ −1 at every iteration. On
the other hand, for every n ≥ 1 the H-components of F n (γ) break
further into pieces as they are mapped onto H-components of F n+1 (γ),
thus decreasing rn (x) at x ∈ γ whose images come close to the new
partition points. We need to show that the expansion prevails over the
partitioning.
The H-components are formed at each iteration of F inductively:
every H-component γ 0 ⊂ F n (γ) is further subdivided into smaller pieces
at points where γ 0 intersects either (i) the singularity manifolds of F or
(ii) the preimages of the boundaries of the homogeneity sections. The
F -images of all those small pieces of all γ 0 ⊂ F n (γ) will constitute the
H-components of F n+1 (γ).
Let us distinguish between the cuttings of type (i) and (ii) here. The
cutting of type (ii) is always accompanied by very strong expansion.
Indeed, if an H-component γ 0 crosses the preimage of the boundary
of a homogeneity section I±k , k ≥ k0 , then a piece γ 00 ⊂ γ 0 will be
mapped into I±k . Now cos ϕ ∼ k −2 on F (γ 00 ), and hence γ 00 undergoes
a strong expansion by a factor ≥ ck 2 for some constant c > 0, due to
Proposition 3.5. This expansion effectively compensates for the cutting
of type (ii), see below.
On the contrary, cutting of type (i), which is caused by the singularities of F , is not necessarily accompanied by strong expansion. One
condition commonly used in the literature to deal with it is the so called
complexity bound
(3.19)
∃n ≥ 1 : Kn θ n < 1
where Kn denotes the maximal number of pieces a small u-curve can
brake into under the map F n . The intuitive meaning of (3.19) is that
a u-curve of a small length δ, after n iteration of the map, will consist
of ≤ Kn pieces of total length ≥ δθ −n , thus the average piece will have
length ≥ δθ −n /Kn > δ.
30
N. CHERNOV AND D. DOLGOPYAT
It appears that the complexity bound (3.19) fails in our setting. Indeed, suppose there is a periodic orbit running back and forth between
two scatterers so that it collides with both scatterers “head-on” (orthogonally). Let γ be a short u-curve moving along this orbit and
experiencing grazing collisions with the slowly moving disk P(Q), see
Fig. 2. Since the disk changes its position between collisions with the
images of γ, it might edge up a bit and cut each smooth component of
the image of γ, roughly in half, at every new collision. Thus, Kn might
grow exponentially fast, and then the complexity bound (3.19) would
fail unless θ happens to be small enough, which we cannot guarantee
in general.
g
Figure 2: A small u-curve that will be cut into 2n pieces by the
moving disk after n runs between the walls.
To save the situation, we observe that the bound (3.19) is rather
crude, because it assumes that H-components only expand with the
minimal possible rate θ −1 . In our case, however, one side of each singularity curve corresponds to a nearly grazing collision, at which the
expansion factor is close to infinity. We will use this fact to prove
Lemma 3.10 (a) without a complexity bound.
We first note that our bound (3.19) is a particular case of Theorem 3.1 in [16] (one only needs to set δ = 0 there). Theorem 3.1
is derived in [16] from three technical assumptions, see (2.6)–(2.8) in
[16]. It suffices to prove analogues of those assumptions in our context.
Except for one important point to be discussed below, the proofs of
(2.6)–(2.8) developed for billiards maps in [16] carry over to our case
without changes, since all the technical components of those proofs
have been prepared by our previous analysis in Sections 3.1–3.3. The
only missing element is the proof of the one-step expansion estimate,
BROWNIAN BROWNIAN MOTION
31
see p. 540 in [16] and (3.20) below, which is proved in [16] with the
help of a complexity bound.
We formulate the one-step expansion estimate in terms of our map F .
Let γ be an H-curve and γi,n all the H-components of F n (γ). For each
−1
i, denote by θi,n
the minimal (local) factor of expansion of the curve
−n
F (γi,n ) under the map F n . Then the one-step expansion estimate
reads
X
(n)
(3.20)
∃n ≥ 1 : α0 := lim inf sup
θi,n < 1
δ→0
γ:|γ|<δ
i
(1)
We will set n = 1 and prove that α0 < 1 without using the complexity bound. This will imply the analogues of (2.6)–(2.8) and hence
Lemma 3.10 (a).
Recall that the singularities of our dynamics are only caused by
grazing (tangential) collisions with the scatterers and the disk P(Q).
At each of them, γ is slices into two parts – one hits the scatterer (or
the disk) and gets reflected (almost tangentially) and the other misses
it (passes by). The reflecting part is further subdivided into countably
many H-components by the boundaries of the homogeneity sections.
An H-component falling into the section Ik or I−k , has been expanded
by a factor ≥ ck 2 , see above. All
make a total
P these H-components
2 −1
contribution to (3.20) less than k≥k0 (ck ) ≤ Const/k0 (note that
the reflecting part of γ lies entirely in ∪k≥k0 I±k provided |γ| is small
enough, which is guaranteed by taking lim inf δ→0 ).
The part passing by without collision may be sliced again by a grazing collision with another scatterer later on, thus creating another
countable set of reflecting H-components. This can happen at most
Lmax /Lmin times, though, where Lmin is the minimal free path of the
light particle, guaranteed by (3.1), and Lmax is the maximal free path of
the light particle (Lmax < ∞ due to our “finite horizon” assumption).
In the end, we will have ≤ Lmax /Lmin countable sets of H-components
resulting from almost grazing collisions and at most one component of
γ that passes by all the grazing collisions and lands somewhere else.
That last component is only guaranteed to expand by a moderate factor
of θ −1 . Thus, we arrive at
(1)
α0 ≤ θ +
Lmax Const
Lmin k0
(1)
Since θ < 1, the required condition α0 < 1 can be enforced by choosing
k0 large enough. This completes the proof of the key estimate (3.20)
without using a complexity bound, and thus yields Lemma 3.10 (a).
32
N. CHERNOV AND D. DOLGOPYAT
Part (b) of Lemma 3.10 follows from (a). Indeed, it suffices to set
β3 = 1/ min{| ln β1 |, | ln θ|}, so that θ n < |γ| and β1n < |γ|, and notice
that mes(x : r0 < εθ n ) < 2eβ̃ εθ n /|γ| due to (3.17).
The proof of (c) is based on a bookkeeping of short H-curves. Pick
ε0 < (1 + β4 )−1 and divide the time interval [n1 , n2 ] into segments of
length s := [2β3 | ln ε0 |]. We will estimate the measure of the set
γ̃ = x ∈ γ : max rn1 +si (x) < ε0
1≤i≤K
where K = (n2 − n1 )/s. For each x ∈ γ̃ define a sequence of natural
numbers S(x) = {k0 , k1 , . . . , km }, with m = m(x) ≤ K, inductively.
Set k0 = 1 and given k0 , . . . , ki we put ti = k0 + · · · + ki and consider
the H-component γi (x) of F n1 +sti (γ) that contains F n1 +sti (x). We set
2k−2
ki+1 = k if |γi (x)| ∈ [ε2k
). If it happens that ti + ki+1 > K,
0 , ε0
we reset ki+1 = K − ti + 1 and put m(x) = i + 1. Note that now
k1 + · · · + km = K.
Next pick a sequence S = {k0 = 1, k1 , . . . , km } of natural numbers
such that k1 + · · · + km = K and let γ̃S = {x ∈ γ̃ : S(x) = S}. We
claim that
mes(γ̃S ) ≤ β4m εK
0
(3.21)
First, by part (b)
(3.22)
mes(k1 (x) = k) ≤ β4 εk0 ,
k≥1
(for k ≥ 2 we actually have mes(k1 (x) = k) ≤ β4 ε2k−2
≤ β4 εk0 ). Then,
0
inductively, for each i = 0, . . . , m − 2 we use our previous notation ti
and γi (x) and put γ̃i (x) = γi (x) if |γi (x)| < 2ε0 , otherwise we denote by
γ̃i (x) ⊂ γi (x) the ε0 -neighborhood of an endpoint of γi (x) that contains
F n1 +sti (x). By Proposition 3.9, the curve γ̃i (x), with the corresponding
conditional measure on it (induced by F n1 +sti (mes` )), makes a standard
pair, call it `i (x). Then again by part (b)
(3.23)
mes`i (x) (ki+2 (x) = k) ≤ β4 εk0 ,
k≥1
(because ki+2 (x) = k implies |γi+1 (x)| < ε2k−2
, and for k = 1 we have
0
|γ̃i+1 (x)| < ε0 ). Multiplying (3.22) and (3.23) for all i = 0, . . . , m − 2
proves (3.21).
Now, adding (3.21) over all possible sequences S = {1, k1 , . . . , km }
gives
K
X
K K
mes(γ̃) ≤
CK−1,m−1 β4m εK
0 = (1 + β4 ) ε0
m=1
(here CK−1,m−1 are binomial coefficients). This proves (c). Lemma 3.10
is proven.
BROWNIAN BROWNIAN MOTION
33
We finish this subsection with a few remarks. Let γ = ∪α γα ⊂ Ω
be a finite or countable union of disjoint H-curves with some smooth
probability measure mesγ on it, whose density of each γα is homogeneous. For every α and x ∈ γα and n ≥ 0 denote by rn (x) the distance
from the point FQn (x) to the nearest endpoint of the H-component of
F n (γα ) to which the point F n (x) belongs. The following is an easily
consequence of Lemma 3.10 (a):
mesγ (x : rn (x) < ε) ≤ (β1 /θ)n mesγ (x : r0 < εθ n ) + β2 ε
Also, there is a constant β7 > 0 such that if mesγ (x : r0 (x) < ε) ≤ β7 ε
for any ε > 0, then mesγ (x : rn (x) < ε) ≤ β7 ε for all n ≥ 1 (it is
enough to set β7 = β2 /(1 − β1 eβ̃ )).
Lastly, suppose γα0 ⊂ γα is a curve, γ 0 = ∪α γα0 , and mesγ 0 is the measure mesγ conditioned on γ 0 . For every α and x ∈ γα0 and n ≥ 0 denote
by rn0 (x) the distance from the point FQn (x) to the nearest endpoint of
the H-component of F n (γα0 ) to which the point F n (x) belongs. It is
easy to see that
mesγ 0 (x : rn0 (x) < ε) ≤
eβ̃
mesγ (x : rn (x) < ε)
mesγ (γ 0 )
Proof of Proposition 2.2 (a) is now a combination of Proposition 3.9 and
Lemma 3.10. We proceed to the equidistribution properties claimed in
Proposition 2.2 (b) and (c).
3.5. Perturbative analysis. Recall that the billiard-type map FQ,V :
ΩQ,V → ΩQ,V is essentially independent of V and can be identified
with FQ : ΩQ → ΩQ by the simple formula πQ,0 ◦ FQ,V = FQ ◦ πQ,0 .
Furthermore, the spaces ΩQ can be identified with the r, ϕ coordinate
space Ω0 by the projection π0 . This gives us a family of billiard maps
FQ acting on the same space Ω0 . They all preserve the same billiard
measure dµ0 = c−1 cos ϕ dr dϕ, where c = 2 length(∂D) + 4πr denotes
the normalizing factor.
We recall a few standard facts from billiard theory [8, 9, 16, 57]. In
the r, ϕ coordinates, u-curves are increasing and s-curves are decreasing, see (3.10) and (3.13), and they are uniformly transversal to each
other. For every Q and integer m, the map FQm is discontinuous on finite union of curves in Ω0 , which are stable for m > 0 and unstable for
m < 0. The discontinuity curves of FQm stretch continuously between
the borders of Ω0 , i.e. from ϕ = −π/2 to ϕ = π/2 (they intersect each
other, of course).
We use the |·|-norm, see Section 3.2, to measure the lengths of stable
and unstable curves. For a u-curve W ⊂ Ω0 and a point x ∈ Ω0 we
34
N. CHERNOV AND D. DOLGOPYAT
define dist(x, W ) to be the minimal length of s-curves connecting x
with W , and vice versa (if there is no such connecting curve, we set
dist(x, W ) = ∞). We define the “Hausdorff distance” between two
u-curves W1 , W2 ⊂ Ω0 to be
dist(W1 , W2 ) = max{ sup dist(x, W2 ), sup dist(y, W1 )}
x∈W1
y∈W2
(and similarly for s-curves). Let W ⊂ Ω0 be a stable or unstable
curve with endpoints x1 and x2 , and ε < |W |/2. For any two points
y1 , y2 ∈ W such that |W (xi , yi )| < ε for i = 1, 2, we call the middle
curve W (y1 , y2 ) an ε-reduction of W .
Next we show that, in a certain sense, the map FQ depends Lipschitz
continuously on Q. Let Q, Q0 satisfy (3.1) and ε = kQ − Q0 k. The
following lemma is a simple geometric observation:
Lemma 3.11. There are constants c2 > c1 > 1 such that
(a) The discontinuity curves of the map FQ0 in Ω0 are within the
c1 ε-distance of those of the map FQ .
(b) Let W ⊂ Ω0 be a u-curve of length > 2c2 ε. Assume that FQ
and FQ0 are smooth on W . Then there are two c2 ε-reductions of
this curve, W̃ and W̃ 0 , such that dist(FQ (W̃ ), FQ0 (W̃ 0 )) < c1 ε.
Corollary 3.12. There is a constant c3 > c2 such that for any H-curve
W ⊂ Ω0 of length > c3 ε there are two finite partitions W = ∪Ii=0 Wi =
∪Ii=0 Wi0 of W such that
(a) |W0 | < c3 ε and |W00 | < c3 ε
(b) For each i = 1, . . . , I, the sets FQ (Wi ) and FQ0 (Wi0 ) are Hcurves such that dist(FQ (Wi ), FQ0 (Wi0 )) < c1 ε.
Proof. The singularities of FQ divide W into ≤ K1 = 1 + Lmax /Lmin
pieces, see the proof of Lemma 3.10 (a). So do the singularities of FQ0 .
Removing pieces shorter than 2c2 ε and using Lemma 3.11 gives us two
partitions W = ∪Jj=0 Ŵj = ∪Jj=0 Ŵj0 such that
(a) |Ŵ0 | < 2K1 c2 ε and |Ŵ00 | < 2K1 c2 ε
(b) For each j = 1, . . . , J, the sets FQ (Ŵj ) and FQ0 (Ŵj0 ) are ucurves such that dist(FQ (Ŵj ), FQ0 (Ŵj0 )) < c1 ε.
Next, for any homogeneity strip Ik and j ≥ 1, consider the H-curves
0
= FQ0 (Ŵj0 )∩Ik . It is easy to see that some
Ŵjk = FQ (Ŵj )∩Ik and Ŵjk
0
, respectively,
Cc1 ε-reductions of these curves, call them Wjk and Wjk
are c1 ε-close to each other in the Hausdorff metric (here C > 0 is the
bound on the slopes of u-curves and s-curves in Ω0 ). Then we take the
0
nonempty curves FQ−1 (Wjk ) and FQ−10 (Wjk
) for all j and k and relabel
BROWNIAN BROWNIAN MOTION
35
them to define the elements of our partitions Wi and Wi0 , respectively.
Using the notation of Lemma 3.10 we have
|W \ ∪i Wi | ≤ 2K1 c2 ε + |{x ∈ W : r1 (x) < Cc1 ε}|
≤ 2K1 c2 ε + β1 θ −1 |{x : r0 (x) < Cc1 θε}| + β2 ε|W |
(we used the estimate in Lemma 3.10 (a), which applies to FQ ). The
resulting bound clearly does not exceed c3 ε for some c3 > 0. A similar
bound holds for |W \ ∪Wi0 |.
Corollary 3.13. There is a constant c4 > 1 such that for each integer
m the discontinuity sets of the maps FQm and FQm0 are c4 ε-close to each
other in the Hausdorff metric.
Proof. We prove this for m > 0 (the case m < 0 is symmetric) by
induction on m. For m = 1 the statement follows from Lemma 3.11
(a). Assume that it holds for some m ≥ 1. Let c4 = 10c2 /(1 − θ).
Now, if the statement fails for m + 1, then there is a point x ∈ Ω0
at which FQm+1
is discontinuous and which lies in the middle of a u0
curve W , |W | = 2c4 ε, on which FQm+1 is smooth. We can assume that
FQ0 is smooth on a c1 ε-reduction Ŵ of W , too, otherwise we apply
Lemma 3.11 (a). Now by Lemma 3.11 (b), there are c2 ε-reductions of
Ŵ , call them W̃ and W̃ 0 , such that dist(FQ (W̃ ), FQ0 (W̃ 0 ) < c1 ε. Due
to our choice of ε4 and the expansion of u-curves by a factor ≥ θ −1 , the
point FQ0 (x) divides FQ0 (W̃ 0 ) into two u-curves of length > c4 ε + 5c2 ε
each. Since FQm0 is discontinuous at FQ0 (x), our inductive assumption
implies that FQm is discontinuous on FQ (W̃ ), a contradiction.
Let Q ∈ D satisfy (3.1) and x ∈ Ω. Denote
(3.24)
εn (x, Q) = max kQ − Q(F i x)k + 1/M
0≤i≤n
where Q(y) denotes the Q-coordinate of y. For a u-curve W ⊂ Ω we put
εn (W, Q) = supx∈W εn (x, Q). Recall that the Q coordinate varies by
< Const/M on u-curves. The following two lemmas are close analogues
of the previous results (with, possibly, different values of the constants
c1 , . . . , c4 ), and can be proved by similar arguments, so we omit details:
Lemma 3.14. For any H-curve W ⊂ Ω of length > c3 ε there are two
finite partitions W = ∪Ii=0 Wi = ∪Ii=0 Wi0 of W such that
(a) |W0 | < c3 ε and |W00 | < c3 ε
(b) For each i = 1, . . . , I, the sets FQ (π0 (Wi )) and π0 (F (Wi0 )) are
H-curves such that dist(FQ (π0 (Wi )), π0 (F (Wi0 )) < c1 ε.
Here ε = ε1 (W, Q).
36
N. CHERNOV AND D. DOLGOPYAT
Lemma 3.15. For any discontinuity point x of the map F n , n ≥ 1,
its projection π0 (x) lies in the c4 ε-neighborhood of some discontinuity
curve of the map FQn , were ε = εn (x, Q).
Lemma 3.16. For any n ≥ 1 the singularity set Sn ⊂ Ω of the map
F n is a finite union of smooth compact manifolds of codimension one
with boundaries. For every Q, V ∈ Υδ1 the manifold Sn intersects ΩQ,V
transversally (in fact, almost orthogonally), and Sn ∩ ΩQ,V is a finite
union of s-curves.
Proof. The first claim follows from our “finite horizon” assumption.
Next, consider the set S̃ ⊂ M of trajectories that experience singularities (grazing collisions) anytime in the future. They make a manifold of
codimension one invariant under the backward dynamics, i.e. Φ−t S̃ ⊂ S̃
for t > 0. Denote by dx∗ = (dQ∗ , dV ∗ , dq ∗ , dv ∗ ) a skew-orthogonal vector to S̃ at a point x = (Q, V, q, v) ∈ S̃ (i.e. such that
hdQ∗ , dQi − hdV ∗ , dV i + hdq ∗ , dqi − hdv ∗ , dvi = 0
for every tangent vector (dQ, dV, dq, dv) ∈ Tx S̃). Since the flow Φt
preserves the symplectic form, the vector DΦ−t (dx∗ ) will be skeworthogonal to S˜ at the point Φt (x) for all t > 0. Note that at the moment immediately preceding of the grazing collision, dQ∗ = dV ∗ = 0
and hdq ∗ , dv ∗ i ≤ −c0 kdq ∗ k kdv ∗ k for some c0 > 0. Hence, by using
the estimates of Section 3.1 we obtain that kdQ∗ k = O(kdq ∗ k/M ),
kdV ∗ k = O(kdv ∗ k/M ) and hdq ∗ , dv ∗ i ≤ −c0 kdq ∗ k kdv ∗ k on the entire
manifold S̃. Thus, skew-orthogonal (i.e., tangent) vectors to Sn ∩ ΩQ,V
are s-vectors, hence it is a union of s-curves.
Now let A be a function from Proposition 2.2. For each point (Q, V )
we define a function AQ,V on Ω0 by
AQ,V = A ◦ (π0 |ΩQ,V )−1
(3.25)
Note that AQ,V has discontinuities on the set SQ,V = π0 (Sn0 ∩ ΩQ,V ).
Also,
Z
Z
Ā(Q, V ) =
A(Q, V, q, w) dµQ,V (q, w) =
AQ,V (r, ϕ) dµ0
ΩQ,V
Ω0
Lemma 3.17. For any (Q, V ) and (Q0 , V 0 ) we have
Ā(Q, V ) − Ā(Q0 , V 0 ) ≤ C kQ − Q0 k + kV − V 0 k
+n0 (kV k + kV 0 k) + n20 /M
for some C = C(A) > 0.
BROWNIAN BROWNIAN MOTION
37
Proof. If A had a bounded local Lipschitz constant (2.8) on the entire space Ω, the estimate would be trivial. However, the function A
is allowed to have singularities on Sn0 (the discontinuity set for the
map F n0 ), and the local Lipschitz constant Lipx A is allowed to grow
near Sn0 , according to Lemma 2.1. As a result, two additional error
terms appear, denoted by E1 + E2 , where E1 comes from the fact that
the functions AQ,V and AQ0 ,V 0 have different singularity sets SQ,V and
SQ0 ,V 0 , and E2 comes from the growing local Lipschitz constant near
these singularity sets.
The error term E1 is bounded by 2kAk∞ Area(G), where G is the
region swept by the singularity set SQ,V as it transforms to SQ0 ,V 0 when
(Q, V ) continuously change to (Q0 , V 0 ). According to Lemma 3.15,
these singularity sets lie within the c4 ε0 -distance from the discontinuity
lines of the maps FQn0 and FQn00 , respectively, where
ε0 = Const n0 (kV k + kV 0 k) + n20 /M
Due to our “finite horizon” assumption, the discontinuity lines of FQn0
and FQn00 have a finite total length, and they lie within the c4 kQ − Q0 kdistance from each other by Corollary 3.13. Obviously, we can cover
G by a finite union of stripes of width 2c4 ε0 + c4 kQ − Q0 k bounded by
s-curves roughly parallel to the discontinuity lines of FQn0 and FQn00 . The
union of these stripes, call it G0 , has area bounded by C(ε0 +kQ−Q0 k),
where C = C(n0 ) > 0, thus
Z
[AQ,V (r, ϕ) − AQ0 ,V 0 (r, ϕ)] dµ0 ≤ 2kAk∞ C(ε0 + kQ − Q0 k)
G0
To estimate the error term E2 , we note that the Lipschitz constants
of the functions AQ,V and AQ0 ,V 0 on the domain Ω0 \ G0 are bounded
by CLA dist(x, G0 )−βA , where x = (r, ϕ) ∈ Ω0 . Here the distance,
originally measured in the Lebesgue metric in Lemma 2.1, can also be
measured in the equivalent | · | metric (the length of the the shortest
u-curve connecting x with ∂G0 ). Thus,
Z
−βA
0
0
E2 ≤ CLA
[ dist(x, G0 )]
dµ0
kQ − Q k + kV − V k
Ω0 \G0
and the integral here is finite because βA < 1.
3.6. Equidistribution properties. We use the following scheme to
estimate E` (A ◦ F n ), where ` = (γ, ρ) is a standard pair, see Proposition 2.2.
38
N. CHERNOV AND D. DOLGOPYAT
Let n1 , n2 (to be chosen later) satisfy K ln |γ| < n1 < n2 < n. For
each point x ∈ γ put
k(x) = min {k : |γk (x)| ≥ ε0 }
n1 <k<n2
where γk (x) denotes the H-component of F k (γ) that contains the point
F k (x) (the constant ε0 was introduced in Lemma 3.10). In other words,
k(x) is the first time, during the time interval (n1 , n2 ), when the image
of the point x belongs in an H-curve of length ≥ ε0 . Clearly, the set
{F k(x) (x) : x ∈ γ} is a union of H-curves of length > ε0 . We denote
those curves by γj , j ≥ 1, and for each γj denote by kj ∈ (n1 , n2 ) the
iteration of F at which this curve was created. Let ρj be the density
of the measure F kj (mes` ) conditioned on γj . Observe that (γj , ρj ) is a
standard pair for every j ≥ 1.
The function k(x) may not be defined on some parts of γ, but by
Lemma 3.10 we have
mes` (x ∈ γ : k(x) is not defined) ≤ Cq n2 −n1
Hence
E` (A ◦ F n ) =
(3.26)
X
j
cj E`j (A ◦ F n−kj ) + O(q n2 −n1 )
where j cj > 1 − Cq n2 −n1 .
We now analyze each standard pair (γj , ρj ) separately, and we drop
the index j for brevity. For example, we denote by mes` = mes`j the
measure on γ = γj with density ρ = ρj . Let x ∈ γ be an arbitrary
point and Q, V its coordinates. For each 0 ≤ i ≤ n − k consider the
map
P
(3.27)
Fi := FQn−k−i ◦ π0 ◦ F i
on the curve γ (here, as in the previous section, we identify the domain
of the map FQ with Ω0 ). Note that Fn−k = π0 ◦ F n−k , and so AQ,V ◦
Fn−k = A◦F n−k , where the function AQ,V on Ω0 was defined by (3.25).
Our further analysis is based on the obvious identity
E` (A ◦ F n−k ) − Ā(Q, V ) = E` (AQ,V ◦ F0 ) − Ā(Q, V )
(3.28)
+
n−k−1
X
i=0
[E` (AQ,V ◦ Fi+1 ) − E` (AQ,V ◦ Fi )]
The following two propositions play a principal role in our argument:
Proposition 3.18. We have
E` (AQ,V ◦ F0 ) − Ā(Q, V ) ≤ Cθ0n−k
for some constants C > 0 and θ0 < 1.
BROWNIAN BROWNIAN MOTION
39
Proof. Since F0 = FQn−k ◦ π0 , our statement just expresses the well
known fact that for dispersing billiards the correlations decay exponentially fast [57, 16]. Actually, the proof of this fact [57] includes the
estimation of convergence of iterations of smooth densities on unstable
curves to the invariant measure, exactly as we claim here. Note that
the u-curve γ has length of order one (|γ| > ε0 ), hence there is no
“waiting period” during which the curve needs to expand – the exponential convergence starts right away. Furthermore, the convergence
is uniform in Q, i.e. C, θ0 are independent of Q, see Extension 1 in
Appendix A.
Proposition 3.19. For each 0 ≤ i ≤ n − k − 1 we have
(3.29)
|E` (AQ,V ◦ Fi+1 ) − E` (AQ,V ◦ Fi )| ≤ Cεγ
where C > 0 is a constant and
εγ : = (n − k)kV k + (n − k)2 /M ≥ max sup kQ − Q(F j x)k
0≤j≤n−k x∈γ
Proof. Estimates of this kind have been obtained for Anosov diffeomorphisms [23] and they are based on shadowing type arguments. We
follow this line of arguments here, too, but face additional problems
when dealing with singularities.
∗
We first outline our proof. We will construct two subsets γi∗ , γi+1
⊂γ
∗
∗
∗
and an absolutely continuous map h : Fi (γi ) → Fi+1 (γi+1 ) (which, in
fact, will be the holonomy map between some H-components of Fi (γ)
and those of Fi+1 (γ)) that have three properties:
∗
(H1) mes` (γ \ γi∗ ) < Cεγ and mes` (γ \ γi+1
) < Cεγ ,
∗
(H2) E`∗i (|AQ,V ◦ Fi − AQ,V ◦ h ◦ Fi |) < Cεγ ,
−1
∗
(H3) the Jacobian J(x) of the map H ∗ : = Fi+1
◦ h∗ ◦ Fi : γi∗ → γi+1
satisfies E`∗i (| ln J|) < Cεγ
R
(here E`∗i (f ) = γ ∗ f dmes` for any function f ). In the rest of the proof
i
of Proposition 3.19, we will denote AQ,V by A for brevity.
Observe that (H1)–(H3) imply (3.29). Indeed, we use the change of
variable y = H ∗ (x) and get
|E` (A ◦ Fi+1 − A ◦ Fi )| ≤ 2kAk∞ Cεγ + E`∗i |(A ◦ h∗ ◦ Fi ) J − A ◦ Fi |
≤ 2kAk∞ Cεγ + Cεγ + E`∗i |(A ◦ h∗ ◦ Fi ) (J − 1)|
≤ 2kAk∞ Cεγ + Cεγ + kAk∞ E`∗i |J − 1|
40
N. CHERNOV AND D. DOLGOPYAT
∗
Next, |E`∗i (J − 1)| = |mes(γi+1
) − mes(γi∗ )| ≤ 2Cεγ , hence
E`∗i |J − 1| ≤ 2Cεγ + E`∗i (1 − J)+
≤ 2Cεγ + E`∗i (| ln J|)
which completes the proof of (3.29) assuming (H1)–(H3).
∗
We begin the construction of the sets γi∗ and γi+1
. First, the definition of the maps Fi and Fi+1 in (3.27) involves the transformation
of the curve γ to F i (γ). Let γ̃ be an H-component of F i (γ). If its
length is < c3 εγ , we simply discard it (i.e., remove its preimage in
∗
γ from the construction of both γi∗ and γi+1
). If |γ̃| > c3 εγ , then
J
Lemma 3.14 gives us two partitions γ̃ = ∪j=0 γ̃i = ∪Jj=0 γ̃j0 , such that
for each j = 1, . . . , J the sets FQ (π0 (γ̃j )) and π0 (F (γ̃j0 )) are H-curves
and dist(FQ (π0 (γ̃j )), π0 (F (γ̃j0 )) < c1 εγ . We remove the preimage of γ̃0
from the construction of γi∗ , and the preimage of γ̃00 from the construc∗
tion of γi+1
. By Lemma 3.10, the total mes` -measure of the (so far)
removed sets is O(εγ ). The remaining H-curves in FQ (π0 (F i (γ)) and
π0 (F i+1 (γ)) are now paired according to Lemma 3.14.
Consider an arbitrary pair of curves W 0 ⊂ FQ (π0 (F i (γ)) and W 00 ⊂
π0 (F i+1 (γ)) constructed above and remember that dist(W 0 , W 00 ) <
c1 εγ . According to our definition of the maps Fi and Fi+1 , both curves
W 0 and W 00 will be then iterated n − k − i − 1 times under the same
billiard map FQ . Let
W∗0 = {x ∈ W 0 : Wxs ∩ W 00 6= ∅}
where Wxs ⊂ ΩQ denotes a homogeneous stable manifold for the map
FQ passing through x. (A homogeneous stable manifold W s ⊂ ΩQ
is a maximal curve such that FQn (W s ) is a homogeneous s-curve for
each n ≥ 0.) Let h : W∗0 → W 00 denote the holonomy map (defined by
sliding along the stable manifolds Wxs ). We remove the preimages of
the points x ∈ W 0 \ W∗0 from the construction of γi∗ , and the preimages
∗
of x ∈ W 00 \ h(W∗0 ) – from the construction of γi+1
.
We need to estimate the measure of set of points that were just
removed. Denote by γ 0 = ∪α γα0 ⊂ Ω0 the union of the above Hcurves W 0 ⊂ FQ (π0 (F i (γ)) and by mesγ 0 the restriction of the measure
FQ (π0 (F i (mes` ))) on γ 0 . For each x ∈ γ 0 and n ≥ 0 denote by rn (x)
the distance from the point FQn (x) to the nearest endpoint of the Hcomponent of FQn (γ 0 ) that contains the point FQn (x). The for any n ≥ 0
and ε > 0
(3.30)
mesγ 0 (x ∈ γ 0 : rn (x) < ε) < βε
where β > 0 is some large constant, according to the remarks in the
end of Section 3.3 (they obviously apply to the billiard map FQ ).
BROWNIAN BROWNIAN MOTION
41
For any x ∈ γ 0 denote by r s (x) the distance from x to the nearest
endpoint of the homogeneous stable manifold Wxs through x. We claim
that for any ε > 0
(3.31)
mesγ 0 (x ∈ γ 0 : r s (x) < ε) < β 0 ε
for some large constant β 0 > 0. Indeed, if r s (x) < ε, then for some
n ≥ 0 the point FQn (x) lies within the (εθ n )-neighborhood of either
a singularity set of the map FQ or the boundary of a homogeneity
strip I±k , k ≥ k0 . Since the singularity lines and the boundaries of
homogeneity strips are uniformly transversal to u-curves, then we have
rn (x) < Cεθ n for some C > 0. Now we use (3.30) to obtain
∞
X
Cβ
Cβεθ n =
mesγ 0 (x ∈ γ 0 : r s (x) < ε) <
ε
1
−
θ
n=0
which proves (3.31).
Now, since the points x ∈ W 0 \ W∗0 and x ∈ W 00 \ h(W∗0 ) removed
∗
from the construction of γi∗ and γi+1
lack stable manifolds of length
c1 εγ , the sets of those points have measure < Const εγ due to (3.31).
∗
This completes the construction of the sets γi∗ and γi+1
and the proof
∗
∗
∗
of (H1). The map h : Fi (γi ) → Fi+1 (γi+1 ) is the induced holonomy
map. It remains to prove (H2) and (H3).
Put d := n − k − i − 1 for brevity. For any point x0 ∈ W∗0 and
its “sister” x00 = h(x) ∈ W 00 , the points z 0 = FQd (x0 ) ∈ Fi (γ) and
z 00 = FQd (x00 ) ∈ Fi+1 (γ) (related by h∗ (z 0 ) = z 00 ) will be (Cθ d εγ )-close,
since FQ contracts stable manifolds by a factor ≤ θ < 1. In other
words, the trajectory of the point x00 shadows that of x0 in the forward dynamics. Therefore, the values of the function A = AQ,V will
differ at the endpoints z 0 and z 00 by at most O(θ d εγ D(z 0 , z 00 )), unless
they are separated by a discontinuity curve of the function A. Here
D(z 0 , z 00 ) = [dist(W s (z 0 , z 00 ), SQ,V )]−βA , where W s (z 0 , z 00 ) denotes the
stable manifold connecting z 0 with z 00 , and SQ,V = π0 (Sn0 ∩ ΩQ,V ) in
accordance with Lemma 2.1 (b).
Let W be an H-component of FQd (W 0 ). Put W∗ = W ∩ FQd (W∗0 )
and mesi = Fi (mes` ). We need to estimate
Z
∆(W ) : =
|A(z 0 ) − A(z 00 )| dmesi
W∗
The curve W crosses the discontinuity set SQ,V in at most K = Kn0
points, cf. Lemma 3.16. The pairs of points z 0 , z 00 separated by discontinuity curves of A lie in a (Cθ d εγ )-neighborhood of those curves, hence
they make a tiny set of | · |-measure < Kn0 Cθ d εγ and their contribution
to ∆(W ) will be ≤ kAk∞ Kn0 Cθ d εγ mesi (W ).
42
N. CHERNOV AND D. DOLGOPYAT
Next, the H-curve W is partitioned by SQ,V into pieces that we
denote by W1 , . . . , Wk for some k ≤ Kn0 . For each piece Wj we put
Wj∗ = Wj ∩ FQd (W∗0 ) and estimate
Z
Z
mesi (Wj ) |Wj | −βA
0
00
0 d
|A(z ) − A(z )| dmesi ≤ C θ εγ
t
dt
|Wj |
Wj ∗
0
≤ C 00 θ d εγ
mesi (Wj )
|Wj |βA
where C 0 , C 00 > 0 are some constants. Summing over j gives
Z
mesi (W )
|A(z 0 ) − A(z 00 )| dmesi ≤ Const Kn0 θ d εγ
|W |βA
W
Next, summing over all the H-components of FQd (W 0 ) and all the curves
W 0 ⊂ FQ (π0 (F i (γ)) gives a bound
X
mesi (W )
∗
d
∗
E`i (|A ◦ Fi − A ◦ h ◦ Fi |) ≤
Const Kn0 θ εγ
|W |βA
W ⊂Fi (γ)
≤ Const Kn0 θ d εγ
where the last bound follows from
Z
X mesi (W )
≤ Const [rn−k (x)]−βA dρ(x) ≤ Const
(3.32)
|β A
|W
γ
W ⊂Fi (γ)
(we note that βA < 1 and use Lemma 3.10). This proves (H2), even
with an additional small factor θ d . It remains to prove (H3).
Let x0− , x00− ∈ γ̃ be the preimages of x0 , x00 , respectively, i.e. x0 =
FQ (π0 (x0− )) and x00 = π0 (F (x00− )). Note that the distance between x0−
and x00− is < Cεγ . Let y 0 = F −i (x0− ) and y 00 = F −i (x00− ) be the preimages
of our two points on the original curve γ. Note that dist(y 0 , y 00 ) ≤ Cθ i εγ
and z 0 = Fi (y 0 ) and z 00 = Fi+1 (y 00 ), see Fig. 3. (We can say that the
trajectory of the point y 00 shadows that of y 0 during all the n − k
−1
iterations.) Now y 00 = H ∗ (y 0 ), where H ∗ = Fi+1
◦ h∗ ◦ Fi . The jacobian
∗
∗
DH of H satisfies
Dγ Fi (y 0 )
ln DH ∗ (y 0 ) = ln
+ ln Dh(z 0 )
Dγ Fi+1 (y 00 )
where Dγ Fi and Dγ Fi+1 denote the jacobian of the maps Fi and Fi+1 ,
respectively, restricted to the curve γ.
Lemma 3.20. We have
d
X
Cεγ
Dγ Fi (y 0 ) Cθ r εγ
≤
+
(3.33)
ln
Dγ Fi+1 (y 00 ) |γ̃|2/3
|γ̃r0 |2/3
r=0
BROWNIAN BROWNIAN MOTION
43
and
∗
0
|ln Dh (z )| ≤
(3.34)
∞
X
Cθ r εγ
r=d
|γ̃r0 |2/3
where γ̃r0 denotes the H-component of FQr+1 (π0 (γ̃)) containing the point
FQr (x0 ).
This lemma is, in a sense, an extension of Proposition 3.6. Its proof
is given in Appendix B, after the proof of Proposition 3.6.
~g
g
F
F
p
p
F
F
Figure 3: The construction in Lemma 3.20
We now complete the proof of (H3):
E`∗i (| ln J|) ≤
X
Cεγ
γ̃⊂F i (γ)
+
∞
X
mes0 (γ̃)
|γ̃|2/3
X
Cθ r εγ
r=0 γ̃ 0 ⊂F r+1 (π0 (F i (γ)))
Q
≤ Cεγ +
∞
X
mes0r (γ̃ 0 )
|γ̃ 0 |2/3
Cθ r εγ = Const εγ
r=0
where mes0 = F i (mes` ) and mes0r = FQr+1 (π0 (F i (mes` ))). Here we used
the same trick as in (3.32). The property (H3) is proved, and so is
Proposition 3.19.
44
N. CHERNOV AND D. DOLGOPYAT
We now return to our main identity (3.28) and obtain
E` (A ◦ F n−k ) − Ā(Q, V ) ≤ Cθ0n−k + C(n − k)εγ
≤ Cθ0n−k + C(n − k)2 kV k + C(n − k)3 /M
Equation (3.26) now yields
X
E` (A ◦ F n ) =
cj Ā(Qj , Vj ) + O(q n2 −n1 )
j
+O(θ0n−n2 ) + O((n − n1 )2 ) max kVj k
γj
3
+O (n − n1 ) /M
where (Qj , Vj ) are the coordinates of an arbitrary point of γj . Note
that
max kVj k ≤ kV k + Cn2 /M
j
Finally, we apply Lemma 3.17 to estimate the value of Ā(Qj , Vj ):
Ā(Qj , Vj ) − Ā(Q, V ) ≤ Cn kV k + Cn2 /M
and arrive at
E` (A ◦ F n ) − Ā(Q, V ) ≤ C kV k n + (n − n1 )2
+ C n2 + (n − n1 )3 /M
+ C q n2 −n1 + C θ0n−n2
Now for any m ≤ min{n/2, K ln M } we can choose n1 , n2 so that n −
n2 = n2 − n1 = m. This completes the proof of Proposition 2.2 (b).
To prove part (c), it suffices to apply part (b) to the last j iterations
of F during the given time interval [0, n].
4. Moment estimates
The main goal of this section is to prove Proposition 2.3 and thus
establish Theorem 2. Our proof is based on various moment estimates
of the underlying processes. In addition, we obtain some more estimates to be used in the proofs of Theorems 1 and 3 presented in the
subsequent sections.
4.1. Plan of the proofs. First we state several propositions that will
lead to the proof of Proposition 2.3. Detailed proofs of our statements
are given in the
√ subsections 4.3–4.7.
Let n = κM M , where for the proof of Theorem 2 we set κM = M −δ
with some δ > 0, whereas for the proof of Theorem 1 we will need κM
to be a small positive constant.
BROWNIAN BROWNIAN MOTION
45
The estimates of Proposition 2.2 require that (Q, V ) ∈ Υδ1 , see (3.1),
so we have to exclude the orbits leaving this region. Fix a δ2 ∈ (δ1 , δ0 )
and let ` = (γ, ρ) be a standard pair such that length(γ) ≥ M −100 . We
define, inductively, subsets
∅ = I0 ⊂ I1 ⊂ . . . Ik ⊂ Ik+1 ⊂ · · · ⊂ γ
as follows. Suppose that
S Ik is already defined so that
(i) F kn (γ \ Ik ) = α γα,k , where for each α we have length(γα,k ) >
M −100 , and `α,k = (γα,k , ρα,k ) is a standard pair (here ρα,k is the density
of the measure F kn (mes` ) conditioned on γα,k );
(ii) π1 (γ \ Ik ) ⊂ Υδ2 , cf. (3.1).
Now, by Proposition 2.2 (a), for each α
!
[
[
F n γα,k =
γα,β,k+1
Rα,k+1
β
where mes`α,k (Rα,k+1 ) < M −50 and length(γα,β,k+1 ) > M −100 for each
β.
We now define
Ik+1 = Ik ∪ ∪α F −(k+1)n (Rα,k+1 ) ∪ ∪∗α,β F −(k+1)n (γα,β,k+1 )
/ Υδ2 . For
where ∪∗α,β is taken over all pairs α, β such that π1 (γα,β,k+1 ) ∈
each x ∈ ` let k(x) = min{k : x ∈ Ik } (we set k(x) = ∞ if x ∈
/ Ik for
any k).
For brevity, for each x ∈ Ω we denote the point F n (x) by xn =
(Qn , Vn , qn , vn ). Given a standard pair ` = (γ, ρ) as above, we define
for every x ∈ γ
Qn for n < kn
Vn for n < kn
(4.1)
Q̂n =
V̂n =
.
Qkn for n ≥ kn
0 for n ≥ kn
Recall that Theorem 2 claims a weak convergence of the stochastic
processes Q(τ ) and V(τ ) on any finite time interval 0 < τ < c. From
now on, we fix c > 0 and set T = c/l, where l is the mean free path
(1.15).
Next, let A ∈ R be a function satisfying two additional requirements:
(a) µQ,V (A) = 0 for all Q, V and (b) µQ,V (Ak ) is a Lipschitz continuous
function of Q and V for k = 2, 3, 4.
Given a standard pair ` = (γ, ρ) as above and x ∈ γ, we put Â(xj ) =
A(xj ) if x 6∈ I[j/n] and Â(xj ) = 0 otherwise, cf. (4.1). Denote Ŝn =
Pn−1
j=0 Â(xj ). In Section 4.4 we will prove
46
N. CHERNOV AND D. DOLGOPYAT
Proposition 4.1. The following bounds hold uniformly for M 1/2 ≤
n ≤ TM 2/3 and for all standard pairs ` = (γ, ρ) such that
(4.2)
π1 (γ) ⊂ Υ∗δ2 ,a := {dist(Q, ∂D) > r + δ2 , kV k < aM −2/3 }
and length(γ) > M −100 :
(a) E` Ŝn = O M δ n/n .
(b) E` Ŝn2 = O(n).
(c) E` Ŝn4 = O(n2 ).
(d) The last estimate can be specified as follows. Let
(4.3)
ωA = max max µQ,V (A2 ), DQ,V (A)
(Q,V )∈Υδ2
where
(4.4)
DQ,V (A) =
∞
X
j=−∞
j
µQ,V A (A ◦ FQ,V
)
Then
E` Ŝn4 ≤ 2ωA2 n2 + O n1.9 .
We remark that part (d) will only be used in Section 6, in the proof
of Theorem 3.
Note that our functions Q̂n , V̂n , Ân and Ŝn are only defined on the
selected standard pair ` and not on the entire phase space Ω yet. Given
an auxiliary measure m ∈ M, we can use the corresponding partition
of Ω into standard pairs {` = (γ, ρ)}, see Section 2.2, and define our
functions on all the pairs {(γ, ρ)} with length(γ) > M −100 , and then
simply set these functions to zero on the shorter standard pairs. Now
our functions are defined on Ω (but they depend on the measure m).
Next given m ∈ M, for each x ∈ Ω we define continuous functions
Q̃(τ ), Ṽ (τ ), and S̃(τ ) on the interval (0, T) by
(4.5) Q̃(τ ) = Q̂τ M 2/3 ,
Ṽ (τ ) = M 2/3 V̂τ M 2/3 ,
S̃(τ ) = M −1/3 Ŝτ M 2/3
(these formulas apply whenever τ M 2/3 ∈ Z, and then we use linear
interpolation in between). In a similar way, let tn be the time of the
nth collision, t̂n = tmin{n,kn} the modified time, and we then define a
continuous function
(4.6)
t̃(τ ) = M −1/3 t̂[τ M 2/3 ] − l min{τ M 2/3 , kn}
BROWNIAN BROWNIAN MOTION
47
where l is the mean free path, cf. (1.15). We note that our normalization factors in (4.5)–(4.6) are chosen so that the resulting functions
typically take values of order one, as we specify next.
Let us fix a > 0 and for each M > 1 choose an auxiliary measure
m ∈ M such that
(4.7)
m π1−1 Υ∗δ2 ,a = 1,
see (4.2). Then each function S̃(τ ), Q̃(τ ), Ṽ (τ ), t̃(τ ) induces a family
of probability measures (parametrized by M ) on the space of continuous functions C[0, T]. Speaking of tightness of Ŝ(τ ), for example, we
mean the tightness of the corresponding family. All our statements
and subsequent estimates will be uniform over the choices of auxiliary
measures m ∈ M satisfying (4.7).
In Section 4.5 we will prove
Proposition 4.2. (a) For every function A ∈ R satisfying the assumptions of Proposition 4.1, the family of functions S̃(τ ) is tight;
(b) the families Q̃(τ ), Ṽ (τ ), and t̃(τ ) are tight.
Corollary 4.3. For every sequence Mk → ∞ there is a subsequence
Mkj → ∞ along which the functions Q̃(τ ) and Ṽ (τ ) on the interval
0 < τ < T weakly converge to some stochastic processes Q̂(τ ) and
V̂(τ ), respectively.
The next step is to use the tightness of Q̂ and V̂ to improve the
estimates of Proposition 4.1. In Section 4.6 we will prove
Proposition 4.4. Let ∆ T be a small positive constant and n =
∆M 2/3 . The following estimates hold uniformly for all standard pairs
` = (γ, ρ) such that π1 (γ) ⊂ Υ∗δ2 ,a and length(γ) > M −100 , and all
(Q̄, V̄ ) ∈ π1 (γ):
1
(a) E` V̂n − V̄ = O
.
M 1/3−δ n
T
2
(A) ∆ M −4/3
(b) E` (V̂n − V̄ )(V̂n − V̄ ) = (1+o∆→0 (1)) σ̄Q̄
(c) E` kV̂n − V̄ k4 = O M −8/3 ∆2
(d) E` Q̂n − Q̄ = (1+o∆→0 (1)) ∆M 2/3 lV̄ +O ∆3/2 ,
(e) E` kQ̂n − Q̄k2 = O ∆2 .
48
N. CHERNOV AND D. DOLGOPYAT
Let us now fix some δ3 ∈ (δ2 , δ0 ). Let B(Q, V ) be a C 3 smooth
function of Q and V with a compact support whose projection on the
Q space lies within the domain dist(Q, ∂D) > r + δ3 . Define a new
function LB on the Q, V space by
2
1X 2
(LB)(Q, V ) = lhV, ∇Q Bi +
σ̄Q (A) ij ∂V2i ,Vj B
2 i,j=1
As before, for each M > 1 we choose a measure m ∈ M satisfying
(4.7). In Section 4.7 we will prove
Proposition 4.5. Let (Q̂, V̂) be a stochastic process that is a limit
point, as M → ∞, of the family of functions (Q̃(τ ), Ṽ (τ )) constructed
above. Then the process
Z τ
M = B(Q̂(τ ), V̂(τ )) −
(LB)(Q̂(s), V̂(s)) ds
0
is a martingale.
Proposition 4.5 implies in virtue of Theorem 4.5.2 of [56] that any
limit process satisfies
dQ̂ = lV̂ dτ,
Q̂(0) = Q0
dV̂ = σ̄Q̂ (A) dw(τ ), V̂(0) = 0
We will need an analogue of the above result for continuous time.
For each x ∈ Ω consider two continuous functions on (0, lT) defined by
Q(τ M 2/3 ) for τ < τ̃
Q̃∗ (τ ) =
Q(τ̃ )
for τ ≥ τ̃
and
Ṽ∗ (τ ) =
where
(4.8)
M 2/3 V (τ M 2/3 ) for τ < τ̃
0
for τ ≥ τ̃
τ̃ = M −2/3 inf{t > 0 : (Q(t), V (t)) ∈
/ Υ δ2 }
In Section 4.7 we will derive
Corollary 4.6. Suppose that (Q̃(τ ), Ṽ (τ )) converges along some subsequence to a process (Q̂(τ ), V̂(τ )). Then (Q̃∗ (τ ), Ṽ∗ (τ )) converges along
the same subsequence to (Q∗ (τ ), V∗ (τ )) = (Q̂(τ /l), V̂(τ /l)).
Corollary 4.6 implies that (Q∗ , V∗ ) satisfies (1.24) up to the moment when dist(Q∗ (τ ), ∂D) = r + δ3 . We can now complete the proof
of Proposition 2.3. Observe that the difference between the limit process Q∗ (τ ) above and the Q(τ ) involved in Theorem 2 is only due to
BROWNIAN BROWNIAN MOTION
49
the different stopping rules (4.8) and (1.23), respectively. In particular, Q∗ can be stopped earlier than Q if for some t ≤ lTM 2/3 we have
M V 2 (t) ≥ 1 − δ2 but dist(Q(s), ∂D) ≥ r + δ0 for all s ≤ t. By Proposition 4.2 (b) the probability of this event vanishes as M → ∞. Thus
any (Q, V) is obtained from the corresponding (Q∗ , V∗ ) by stopping
the trajectory of the latter as soon as dist(Q∗ (τ ), ∂D) = r + δ0 . This
fact concludes the proof of Proposition 2.3 and that of Theorem 2. 4.2. Structure of the proofs. After having completed the formal
description of the intermediate step let us give an informal overview of
the proof.
The proof is based on the martingale method of Stroock and Varadhan [56] which requires the estimates of the first two moments of
Vm − V0 . Hence the key step in the proof of Proposition 2.3 is Proposition 4.4, especially part (b) saying that
2
(4.9)
E` hVn − V0 , ui2 ∼ nhσ̄Q
(A)u, ui/M 2
0
for n ∼ ∆M 2/3 (see the discussion in Section 1 for the motivation of
this identity). The proof of (4.9) proceeds in two steps. First we show
that
n−1
1 X 2
2
hσ̄ (A)u, ui
(4.10)
E` hVn − V0 , ui ∼ 2
M j=0 Qj
(cf. Lemma 4.12). Next we prove that
(4.11)
Q j ∼ Q0
(cf. Proposition 4.2).
For any fixed j, (4.11) follows from
(4.12)
E` kVn − V0 k2 ≤ Const n/M 2
by the Cauchy–Schwartz inequality. In turn (4.12) follows from (4.10).
Hence the key step is to establish (4.10). (Note that in order to get
(4.11) for all j, we need to control the fourth moment, but the necessary
fourth moment estimate follows from (4.10) with little difficulty.)
The proof of (4.10) is based on Proposition 2.2. For small n, part
(b) suffices. However, for n ∼ M 2/3 the term nkV̄ k becomes of order
1, so part (b) is too crude. We first establish (4.10) for n ∼ jM 1/2−δ
(Section 4.3) and then derive it for n ∼ jM 1/2−δ using part (c) and the
inductive estimate
p
(j − 1)M 1/2−δ
E` ||V(j−1)M 1/2−δ || ≤ Const
M
(this is done in Section 4.4).
50
N. CHERNOV AND D. DOLGOPYAT
Finally let us comment on the above inductive step. Let A(u) =
hA, ui. Then (4.10) states that the main contribution to

!2 
n
n
X
X
(u)
E` (A(u) (xi )A(u) (xj )
(4.13)
E` 
A (xj )  =
j=1
i,j=1
comes from nearly diagonal terms (where i ≈ j). To prove this, we
need to bound the contribution of the off-diagonal terms. There are
two mechanisms to get cancellations in (4.13):
(I) Use Proposition 2.2 (c) to estimate E` (A(u) (xi )A(u) (xj ) . Since
we expect the change of V to be of order M −2/3
estimate
, the best−2/3
(u)
(u)
we can get in this way is E` (A (xi )A (xj ) = O(M
ln M ).
2/3 2
Since there are [M ] terms, this provides the off-diagonal bound
of M 2/3 ln M , which is even larger than the main term.
(II) For a fixed i, we can try to get the inductive bound
!
p
X
(u)
(u)
E`
A (xj )A (xi ) ≤ O
# of terms = O(M 1/3 ).
j
This would give the off-diagonal bound of O(M 2/3+1/3 ) = O(M ), which
is even worse.
To overcome these difficulties, we use the so called big-small block
method. Namely, we divide the interval [1, n] into blocks of size M 1/2−δ
separated by small blocks of size M δ . Then the contribution of the small
blocks is negligible. On the other hand, denoting by Pj0 the contribution
P
of the jth big block and setting Uk0 = kj=1 Pj0 we will show that
0 2
0 2
0
E` (Uk+1
) − (Uk0 )2 = E` (Pk+1
) + 2 E` Uk0 Pk+1
.
The first term here can be handled by the method (I) while for the
cross-product term we get, by Proposition 2.2,
0
E` (Uk0 Pk+1
) ≤ Const E` (|Uk0 |) E(Pk0 ).
√
Now the first factor is of order kM 1/2+δ by the method (II), while
the second factor is of order 1 by the method (I). Hence the big-small
block technique allows us to get the optimal bound by combining the
two approaches (I) and (II).
4.3. Short term moment estimates for V . Here we estimate
√ the
moments of the velocity V during time intervals of length n = O( M ),
which are much shorter than O(M 2/3 ) required for Theorem 2. Our
estimates will be used later in the proof of Proposition 4.1. The main
result of this subsection is
BROWNIAN BROWNIAN MOTION
51
Proposition 4.7. Let ` = (γ, ρ) be a standard pair such that π1 (γ) ⊂
Υδ1 and length(γ) > M −100 . Then for all (Q̄, V̄ ) ∈ π1 (γ) and M 1/3 ≤
n ≤ cM 1/2 we have
(a) E` (Vn −V̄ ) = O M δ−1
(b) E` (kVn −V̄ k2 ) = O n/M 2 .
Here c δ1 is the constant of Proposition 2.2.
Proof. Let ∆Vj = Vj+1 − Vj . Then by (1.8)
1
A ◦ Fj
+O
(4.14)
∆Vj =
M
M 3/2
where A is as defined in Section 4.1 Hence
n−1
X
−1
Vn − V̄ = M
A ◦ F i + O nM −3/2 } .
i=0
Also,
2
n−1
X
i
2
−2 A ◦ F + O n2 M −3 .
kVn − V̄ k ≤ 2M i=0
Therefore Proposition 4.7 follows from the next result:
Proposition 4.8. Let A ∈ R be a function satisfying µQ,V (A) = 0 for
Pn−1
all Q, V . Let `, n be as in Proposition 4.7 and Sn = i=0
A ◦ F i . Then
(a) E` (Sn ) = O M δ
(b) E` (Sn2 ) = O(n).
The proof uses the so called big small block techniques [3]. For each
k = 0, . . . , [n/M 1/3 ] denote
(k+1)M 1/3
Rk0
X
=
A(xj ),
j=kM 1/3 +M δ
Rk00
=
kM 1/3
+M δ −1
X
A(xj ),
j=kM 1/3
Zk0
=
k−1
X
j=0
Rj0 ,
Zk00
=
k−1
X
Rj00 .
j=0
Observe that Zk00 ≤ kAk∞ n/M 1/3−δ . Next we prove two lemmas:
52
N. CHERNOV AND D. DOLGOPYAT
Lemma 4.9. For every k,
(a) E` (Rk0 ) = O M 1/3+δ kV̄ k + n/M
(b) E` ([Rk0 ]2 ) = O M 1/3 .
.
Lemma 4.10.
0
(a) E` (Zk+1
) = E` (Zk0 )+O M 1/3+δ kV̄ k + n/M .
√
0
(b) E` ([Zk+1
]2 ) = E` ([Zk0 ]2 )+O M 1/3 +O
k + 1M 1/2+δ kV̄ k + n/M ,
E` ([Zk0 ]2 ) ≤ DM 1/3 (k + 1).
Proof of Lemma 4.9. Applying Proposition 2.2(c) to m ≤ n iterations of F , setting j = M δ/4 , and using the obvious bound
we get
kVm−j k ≤ kV̄ k + Const m/M
E` (A(xm )) = O
kV̄ k + m/M M δ/2 + O M 3δ/4−1
Now (a) follows by summation over m = kM 1/3 + M δ , . . . , (k + 1)M 1/3 .
To prove (b) we write
X
X
(4.15)
(Rk0 )2 =
A(xi )A(xj ) = 2
A(xi )A(xj ) + O M 1/3 .
i<j
i,j
Thus it suffices to show that
(4.16)
j−i
|E` (A(xi )A(xj ))| < Const θA
+ kV̄ k + n/M M δ
for some θA < 1. To prove (4.16) we apply Proposition 2.2 (a) with
n = (i + j)/2. Denoting m = (j − i)/2 we obtain
X
E` (A(xi )A(xj )) =
cα E`α (A(x−m )A(xm )) .
α
If length(γα ) > exp(−m/K) where K is the constant of Proposition 2.2,
choose x̄α ∈ γα . Due to (2.10) and the Hölder continuity of A, for any
m
xα ∈ γα we have |A(F −m xα ) − A(F −m x̄α )| = O(θA
), therefore
(4.17)
m
E`α (A(x−m )A(xm )) = A(F −m x̄α ) E`α (A(xm )) + O(θA
).
By the argument used in the proof of Lemma 4.9 (a)
E`α A(xm ) = O (kV̄ k + m/M )M δ/2 ,
hence
m
+ kV̄ k + m/M M δ .
E`α (A(x−m )A(xm )) = O θA
On the other hand, the contribution of α’s with length(γα ) ≤ exp(−m/K)
is exponentially small due to Proposition 2.2 (a). Summation over α
BROWNIAN BROWNIAN MOTION
53
gives (4.16).
√ Lastly, performing
√ summation over i, j and remembering
that n ≤ c M and kV̄ k < 1/ M we get Lemma 4.9 (b).
Proof of Lemma 4.10. The part (a) follows directly from Lemma 4.9
(a).
To prove part (b) we expand
(4.18)
0
0
0
E` ([Zk+1
]2 ) = E` ([Zk0 ]2 ) + E` ([Rk+1
]2 ) + E` (Zk0 Rk+1
).
The second term is O(M 1/3 ) by Lemma 4.9 (b). We will show that the
last term is much smaller, precisely
1 0 0
(4.19)
E` (Zk Rk+1 ) = O M 12 +δ
The argument used in the proof of (4.16) gives
δ
M
|E` (Zk0 Aj )| ≤ Const E` |Zk0 | θA
+ kV̄ k + n/M M δ
for (k + 1)M 1/3 ≤ j ≤ (k + 2)M 1/3 . Hence
δ
0
M
|E` (Zk0 Rk+1
)| ≤ Const E` |Zk0 | θA
+ kV̄ k + n/M M δ M 1/3
q
≤ Const E` ((Zk0 )2 ) kV̄ k + n/M M 1/3+δ .
(where we used the Cauchy-Schwarz inequality). By induction
q
√
0 0
E` (Zk Rk+1 ) ≤ C D
(k + 1)M 1/3 kV̄ k + n/M M 1/3+δ .
Since k ≤ cM 1/6 , the the right hand is O(M 1/12+δ ). If D is sufficiently
large, this implies both inequalities of part (b) for k + 1 and thus
completes the proof of Lemma 4.10.
Proof of Proposition 4.8. To simplify our analysis we assume that
n = kM 1/3 for some integer k, so that Sn = Zk0 + Zk00 . Similarly to the
proof of Lemma 4.9 (a) we get
E` (Rj00 ) = O M δ (kV̄ k + n/M )
for all 1 ≤ j ≤ k − 1 and
E` (R000 ) = O M δ .
(The difference between the first term and the other terms here is due
to the restriction n > K| ln length(γ)| in Proposition 2.2.) Combining
the above estimates with Lemma 4.9 (a) we obtain part (a). To prove
part (b) we estimate
E` Sn2 ≤ 2 E` [Zk0 ]2 + 2 E` [Zk00 ]2 = O(n + k 2 M 2δ ).
This completes the proof of Proposition 4.8 and hence 4.7.
54
N. CHERNOV AND D. DOLGOPYAT
4.4. Moment estimates–a priori bounds. Here we prove Proposition 4.1.
First we obtain a useful lemma, which shows that the future and the
past dynamics are asymptotically independent. Let A1 , . . . , Ap and
B1 , . . . , Bq be some functions from our class R and c1 , c2 some constants. Consider the functions
X
X
A(x) =
A1 (xi1 ) · · · Ap (xip )−c1 , B(x) =
B1 (xj1 ) · · · Bq (xjq )−c2
∗
∗∗
P
P
where the summations ∗ and ∗∗ are performed over two different
sets of indices (time moments). Let m∗ be the maximal index in the
first set (denoted by ∗) and m∗∗ the minimal index in the second set.
We suppose that m∗ ≤ m − M δ < m ≤ m∗∗ for some m, i.e. there is a
“time gap” of length ≥ M δ between m∗ and m∗∗ .
Now, let ` = (γ, ρ) be a standard pair such that length(γ) > M −100 .
We can decompose the expectation
X
1
δ
(4.20)
E` (· ◦ F m− 2 M ) =
E`α (·)
α
1
δ
where `α denote the components of the image of ` under F m− 2 M .
Lemma 4.11 (“independence”). We have
|E` (A(x)B(x))| ≤ E` |A(x)| max E`α B(x−m+ 1 M δ ) + O M −50
α
2
where the maximum is taken over α’s in (4.20) with length(γα ) >
M −100 . We note that the remainder term O (M −50 ) here depends on
the choice of the functions Ai , Bj and the constants c1 , c2 .
The proof of Lemma 4.11 is similar to that of (4.16), in which the
factorization (4.17) plays a key role.
We now turn to the proof of Proposition 4.1. Using the big small
blocks again, we put for all k = 0, . . . , [TM 2/3 /n]
j=(k+1)n
Pk0
=
X
Â(xj ),
Pk00
=
δ
kn+M
X
j=kn
j=kn+M δ
and then
Uk0
=
k−1
X
Pj0 ,
j=0
Note that Uk00 = O (k + 1)M
δ
Uk00
=
k−1
X
j=0
.
Pj00 .
Â(xj ),
BROWNIAN BROWNIAN MOTION
55
Lemma 4.12. Under the assumptions of Proposition 4.1 we have, uniformly in k,
(a) E` (Pk0 ) = O M δ .
0 2
(b) E` [Pk ] = (1 + g) n E` D̂Qkn ,Vkn
where
2
D̂Q
(x)
kn ,Vkn
=
and g → 0 as M → ∞ and κM
(c) E` [Pk0 ]4 = O n2 .
2
DQ
(A) if x ∈
/ Ik
kn ,Vkn
0
otherwise
→ 0, see a remark below.
(d) In the notation of (4.3), we have
E` [Pk0 ]4 ≤ 2ωA n2 + O n1.9
.
Recall that in the proof of Theorem 2 we set κM = M −δ , hence
κM → 0 follows from M → ∞, and so we can replace g in (b) by o(1).
In the proof of Theorem 1, however, κM will be a small constant, hence
the condition κM → 0 will be necessary.
Proof. We first note that  is different from 0 only on standard pairs
where (2.9) holds, see the construction of  in Section 4.1. Thus,
Proposition 2.2 applies to each standard pair γα,k where  6= 0. Therefore it is enough to verify Lemma 4.12 for k = 0 (but we need to
establish it for all ` = (γ, ρ) such that π1 (γ) ⊂ Υδ2 ).
Part (a) follows from Proposition 4.8 (a).
The proof of (b) is based on the following claim: for each ε > 0 there
exists K(ε) (it is enough to set K(ε) = Const | ln ε|) such that


X
(4.21)
E` 
Â(xi )Â(xj ) < εn
|i−j|>K(ε)
(here, of course, M δ ≤ i, j ≤ n). To prove (4.21), we apply the big
small block decomposition, as in Lemmas 4.9 and 4.10, to [Pk0 ]2 (with
big blocks of length M 1/3 and small blocks of length M δ ), then we use
Eqs. (4.18)–(4.19), the induction on k, and the estimate (4.16) applied
to for each big block will yield (4.21).
Therefore, to get the asymptotics of E` ([P00 ]2 ) we need to get the
asymptotics of
!
n
X
Â(xi )Â(xi+m )
E`
i=M δ
56
N. CHERNOV AND D. DOLGOPYAT
for each fixed m. Applying Proposition 2.2 (a) to j = i − M δ iterations
of F we get
X
E` (Â(xi )Â(xi+m )) =
cα E`α Â(xM δ )Â(xM δ +m ) .
α
where `α = (γα , ρα ) denote the components of the image of ` at time j.
Proposition 2.2 applies to each γα where Â(xM δ ) 6= 0, and by the part
(b) of that proposition, for each α such that length(γα ) > exp(−M δ /K)
we have
E`α Â(xi )Â(xi+m ) = µQ̄,V̄ Â(x0 )Â(xm ) + O kV̄ kM δ + M 2δ−1
where (Q̄, V̄ ) ∈ π1 (γα ) is an arbitrary point. As before, the contribution of small γα is well within the error bounds of our claim (b), hence
summing over α and using the fact that the oscillations of Q and V
over γα are of order 1/M gives
E` Â(xi )Â(xi+m ) = E` µQj ,Vj (Â(x0 )Â(xm )) +O E` (kV̂j k)M δ + M 2δ−1 .
(recall that j = i − M δ ). By Proposition 4.7
√
E` kV̂j k = O kV̄ k + i/M = O kV̄ k + M −3/4
whereas by Lemma 3.17
m
E` µQj ,Vj Â(x0 )Â(xm )
= E` µQ̄,V̄ Â (Â ◦ FQ̄,V̄ ) + O E` kQ̂j − Q̄k
+O E` kV̂j − V̄ k + O kV̄ k + M −1
Proposition 4.7 (b) gives
p
E` kV̂j − V̄ k ≤ Const j/M ≤ Const M −3/4
(we note that n = κM M 1/2 < cM 1/2 , hence Proposition 4.7 indeed
applies in our context). Also, since M kVj k2 < 1 − δ1 , then kvj k ≥
Const > 0, hence intercollision times are uniformly bounded above for
all j ≤ kn. Therefore,
E` kQ̂j − Q̄k
≤ Const
j−1
X
p=0
E` kV̂p k
≤ Const j kV̄ k +
j−1
X
p=0
E` kV̂p − V̄ k
≤ Const j kV̄ k + j M −3/4 .
BROWNIAN BROWNIAN MOTION
57
Note that
√
j kV̄ k ≤ Const n/ M = Const κM ,
j M −3/4 ≤ κM M −1/4 .
This gives
E` Â(xi )Â(xi+m ) = µQ̄,V̄ Â (Â ◦
m
FQ̄,
V̄ )
+ O (κM ) .
We note that all our constants and the O(·) terms depend, implicitly,
on m which takes values between 0 and Kε . Summing over i, m we get


X
m
,
E` [P00 ]2 = n 
µQ̄,V̄ Â (Â ◦ FQ̄,
V̄ ) + χ + O (κM )
|m|<K(ε)
where |χ| ≤ ε and the O(·) term implicitly depends on ε. Now for any
ε > 0 we can choose a small enough κM so that the |O(κM )| < ε. This
concludes the proof of part (b).
We proceed to the proof of part (d). We write
X
E` [P00 ]4 =
E` Â(xi1 )Â(xi2 )Â(xi3 )Â(xi4 ) .
i1 ,i2 ,i3 ,i4
For convenience, we order the indices in each term so that
i 1 ≤ i2 ≤ i3 ≤ i4
(4.22)
There are eight cases depending on the choice of “<” or “=” in (4.22),
however we will be able to handle several cases together. First we
separate the terms in which i1 ≤ i2 < i3 < i4 and get
!
n
n
X
X
2
(4.23)
E` [P00 ]4 =
E` Ŝm
Â(xm )
Â(xj ) + R
j=m+1
m=M δ +1
where we denote, for convenience, j = i4 , m = i3 and Ŝm =
while R correspond to the remaining terms. Denote
Ŝ (a) = Ŝm−M δ
then we have
Ŝ (c) = Ŝm+M δ − Ŝm
2
E` Ŝm
Â(xm )
X
Â(xj )
j>m
+E`
Ŝ (b) = Ŝm − Ŝm−M δ ,
[Ŝ (b) ]2 Â(xm )
X
j>m
!
Â(xj )
i=M δ
Â(xi ),
Ŝ (d) = Ŝn − Ŝm+M δ ,
= E`
!
Pm−1
[Ŝ (a) ]2 Â(xm )
X
Â(xj )
j>m
+ 2E`
Ŝ (a) Ŝ (b) Â(xm )
= I + II + III.
X
j>m
!
Â(xj )
!
58
N. CHERNOV AND D. DOLGOPYAT
We can assume here that m > M 3δ , because terms with m < M 3δ make
a total contribution of order nM 9δ , which is small. It is clear that
part (b) of Lemma 4.12 can
to any number of iterations
be applied
3δ
(a) 2
M < m ≤ n, hence E` [Ŝ ]
< (ωA + o(1)) m. Therefore by
Lemma 4.11
!
n−m
X
|I| < m(ωA +o(1)) max E`α A(xM δ /2 )
A(xj+M δ /2 ) +O M −50 ,
α j=1
where `α denote the components of the image of ` at time m − M δ /2.
Similar to the proof of Lemma 4.12 (b), for each α we have
(4.24)
!
∞
n−m
X
X
j
µQ̄,V̄ A (A ◦ FQ̄,V̄ ) + o(1).
Â(xj+M δ /2 ) ≤
E`α Â(xM δ /2 )
j=1
j=1
We observe that
∞
D (A) − µ (A2 )
X
Q̄,V̄
Q̄,V̄
j
)
=
µQ̄,V̄ A (A ◦ FQ̄,
≤ ωA ,
V̄
2
j=1
hence I = O (ωA2 m) . Next,
(a) (b)
(c)
(d)
III = E` Ŝ Ŝ Â(xm )[Ŝ + Ŝ ]
= E` Ŝ (a) Ŝ (b) Â(xm )Ŝ (c) + E` Ŝ (a) Ŝ (b) Â(xm )Ŝ (d)
= III c + III d .
Now by Lemma 4.11
√
|III c | ≤ Const M 2δ E` S (a) ≤ Const M 2δ m
where the last inequality is based on Lemma 4.12 (b) and CauchySchwarz. On the other hand, due to Lemma 4.11 and Proposition 4.7
(a)
|III d | ≤ Const kV̄ k + M δ−1 E` Ŝ (a) Ŝ (b) Â(xm )
≤ Const M 2δ−1 E` (|Ŝ (a) |)
√
≤ Const M 2δ−1 m
(here again the last
√ inequality follows from Lemma 4.12 (b)). Thus
|III| ≤ Const M 2δ m. Similar estimates show that |II| ≤ Const M 3δ .
Combining these results we get
!
X
√
2
Â(xj ) ≤ 2ωA2 m + O M 2δ m + M 3δ .
E` Ŝm Â(xm )
j>m
BROWNIAN BROWNIAN MOTION
59
Summation over m gives
!
X
X
2
E
Ŝ
Â(x
)
Â(x
)
`
≤ 2ωA2 n2 + O n3/2+6δ .
m
j
m
m
j>m
It remains to estimate the term R in (4.23), which corresponds to the
cases where i2 = i3 or i3 = i4 . The cases where i1 ≤ i2 < i3 = i4 can
be treated in the same way as above, except (4.24) now takes form
E`α Â2 (xM δ /2 ) = µQ̄,V̄ (A2 ) + o(1) ≤ ωA
and the term III d is missing altogether.
The case i1 = i2 = i3 < i4 and that of i1 < i2 = i3 = i4 can be
handled as follows:
!
X X
Â(xk )3 Â(xi ) ≤ Const
E |Ŝn | + 1 ≤ Const n3/2 .
E `
k
i6=k
Consider the case i1 < i2 = i3 < i4 . Using the same notation as in the
analysis of the first term in (4.23) we get
X X I ac + I ad + I bc + I bd
=
E` Ŝ (a) + Ŝ (b) Â2 (xm ) Ŝ (c) + Ŝ (d)
m
m
where we denoted I αβ = E` Ŝ (α) Ŝ (β) Â2 (xm ) . The estimation of each
term here is similar the ones discussed above, and we obtain
q
√
ad
δ
−50
(a)
2
I =O
E` (Ŝ ) M + M
= O m Mδ .
√
m Mδ .
I bc = O M 2δ .
I bd = O M 2δ .
I ac = O
Hence
X
i1 <m<i4
E` Â(xi1 )Â2 (xm )Â(xi4 ) = O n3/2 .
We have just one case i1 = i2 = i3 = i4 left, which is simple:
X 4
E` Â (xj ) ≤ Const n.
j
This proves part (d). Obviously, part (c) follows from (d). This completes the proof of Lemma 4.12.
We now return to the proof of Proposition 4.1. Denote
(4.25)
m(. . . ) = max |E` (. . . )|
`
60
N. CHERNOV AND D. DOLGOPYAT
where the maximum is taken over all standard pairs ` = (γ, ρ) with
length(γ) > M −100 .
We are now going to prove by induction that
(4.26)
m(Uk0 ) ≤ G1 kM δ ,
(4.27)
m([Uk0 ]2 ) ≤ G2 kn,
(4.28)
m([Uk0 ]4 ) ≤ G4 k 2 n2
provided the constants G1 , G2 , G4 are sufficiently large. Let us rewrite
the estimates of Lemma 4.12 in a simplified way:
(4.29) E` (Pk0 ) ≤ C1 M δ , E` [Pk0 ]2 ≤ C2 n, E` [Pk0 ]4 ≤ C4 n2 .
Now, by the inductive assumption (4.26) we have
E ` U 0
≤ G1 kM δ + C1 M δ ,
k+1
hence (4.26) holds for k + 1 provided that G1 > C1 .
Next, by Lemma 4.11, (4.29), the inductive assumption (4.27), and
the Cauchy-Schwarz inequality we have
0
m [Uk+1
]2 ≤ m [Uk0 ]2 + 2 m (Uk0 Pk0 ) + m [Pk0 ]2
p
≤ G2 kn + 2 C1 M δ G2 kn + C2 n.
(4.30)
Since kn < TM 2/3 , the second term here is O(M −1/6+2δ n), hence (4.27)
holds provided that G2 > C2 .
Lastly, by the inductive assumption (4.28) we have
0
m [Uk+1
]4 ≤ m [Uk0 ]4 + 4m [Uk0 ]3 Pk0 + 6m (Uk0 )2 (Pk0 )2
+4m Uk0 [Pk0 ]3 + m [Pk0 ]4
≤ G4 k 2 n2 + I + II + III + IV .
Using Lemma 4.11, (4.29), and the Hölder inequality we get
r
k δ
3/4
3/4
M ,
I ≤ 4C1 G4 [kn]3/2 M δ = 4C1 G4 kn2
n
II ≤ 6(G2 kn) × (C2 n) = 6(G2 C2 )kn2 ,
p
p
√ 2
3/4
3/4
III ≤ 4( G2 kn) × (C4 n3/2 ) = G2 C4
kn ,
and
IV ≤ C4 n2 .
Hence, (4.28) holds provided that G4 > 6C2 G2 . This completes the
proof of (4.26)–(4.28) establishing the parts (a)–(c) of Proposition 4.1
(the contribution from Uk00 is well within our error bounds). The proof
of (d) is similar to (c) but we have to use Lemma 4.12 (d) in place of
Lemma 4.12 (c). Proposition 4.1 is proved.
BROWNIAN BROWNIAN MOTION
61
Let us also note, for future
that by (4.30) the main dif references,
0
2
0 2
ference between E` [Uk+1 ] and E` ([Uk ] ) comes from the [Pk0 ]2 term.
Hence we have
X
2
(4.31)
E` ŜTM
=
E` [Pk0 ]2 + O M 1/3+2δ .
2/3
k≤TM 2/3 /n
4.5. Tightness. We start the proof of Proposition 4.2 by a few general
remarks.
To establish the tightness of a family of probability measures {PM }
on the space of continuous functions C[0, T] we need to show that
for any ε > 0 there exists a compact subset Kε ⊂ C[0, T] such that
PM (Kε ) > 1−ε for all M . The compactness of Kε means that the functions {F ∈ Kε } are uniformly bounded at τ = 0 and equicontinuous
on [0, T]. All our families of functions in Proposition 4.2 are obviously uniformly bounded at zero, hence we only need to worry about
the equicontinuity. Next, for any M0 > 0 our functions S̃(τ ), Q̃(τ ),
Ṽ (τ ), and t̃(τ ) that correspond to M < M0 have uniformly bounded
derivatives (with a bound depending on M0 ), hence they make a compact set. Thus it is enough to construct a compact set Kε such that
PM (Kε ) > 1 − ε for all M > Mε with some Mε > 1, hence we can (and
will) assume that M is large enough.
Lastly, recall that each m ∈ M satisfies (2.3)–(2.5) and now we
can assume that CMε−50 < ε/2. Thus, it will be enough to prove all
necessary measure estimates for measures mes` on individual standard
pairs ` = (γ, ρ) with length(γ) > M −100 (but our estimates must be
uniform over all such standard pairs).
First we prove the part (a) of Proposition 4.2. Let CN be the space
of continuous functions S(τ ) on [0, T] such that
k
+
1
k −m
S
8
(4.32)
≤
2
−
S
2m
2m for all m ≥ N and k < 2m T. We claim that for each > 0 there exists
N such that for all ` with length(γ) > M −100
mes` (S̃ ∈ CN ) > 1 − uniformly in M. Note that by (4.5)
k
+
1
k kAk∞ M 1/3
S̃
(4.33)
≤
−
S̃
2m
2m 2m
so (4.32) is automatic if 2−m < Const M −8/21 . Hence we can assume
that 2−m ≥ Const M −8/21 . Equivalently, we need to estimate |Ŝn2 − Ŝn1 |
for |n2 − n1 | ≥ Const M 2/7 .
62
N. CHERNOV AND D. DOLGOPYAT
Lemma 4.13. For all n1 , n2 such that |n2 − n1 | > Const M 2/7 and for
all ` with length(γ) > M −100
h
i4 ≤ Const (n2 − n1 )2 .
E` Ŝn2 − Ŝn1
Proof. For n2 − n1 ≥ cM 1/2 , our estimate follows from Lemma 4.12 (c)
and the argument used in the proof of Proposition 4.1 (c). For smaller
n2 − n1 , the proof is similar to that of Lemma 4.12 (c).
Lemma 4.13 implies that for fixed k, M
k+1
k −m/8
mes` S̃
− S̃
>2
=
2m
2m !
4
k
+
1
k
> 2−m/2 ≤
mes` S̃
− S̃
2m
2m 4 !
k
+
1
k 2m/2 E` S̃
−
S̃
≤
2m
2m 2m/2
= Const 2−3m/2 .
2m
2
Summation over k and m completes the proof of part (a) of Proposition
4.2.
Next we prove part (b). The tightness of Ṽ (τ ) follows from (1.8),
(4.14) and Proposition 4.2 (a) applied to the function A (the contribution of the correction term O M −3/2 in (4.14) is well within our error
bounds).
The equicontinuity of Q̃(τ ) follows from a simple estimate:
Const
kQ̃(τ2 ) − Q̃(τ1 )k ≤ (τ2 − τ1 ) max kṼ (τ )k
τ
Hence the function Q̃(τ ) is Lipschitz continuous with Lipschitz constant
max[0,T ] kṼ (τ )k2 that can be bounded by the tightness of Ṽ (τ ). Thus
the tightness of Q̃(τ ).
To prove the tightness of t̃(τ ) we consider intercollision times ξj =
t̂j+1 − t̂j = dˆj /kvj k, where dj is the distance between the points of the
jth and (j + 1)st collisions and dˆj = dj 1j≤kn , for all 0 ≤ j ≤ M 2/3 T.
Note that kvj k ≥ Const > 0 for all j < kn, hence ξj ≤ Const. Let ˆlj
equal l if j < kn and 0 otherwise.
Consider the function d(x), x ∈ Ω, equal to the distance between
the positions of the light particle at the points x and F (x) (the distance between its successive collisions). In Appendix A we prove the
following:
BROWNIAN BROWNIAN MOTION
63
Proposition 4.14. The function d belongs in our space R. The average µQ,V (dk ) is a Lipschitz continuous function of Q, V for k ∈ N. In
particular, we have
µQ,V (d) = l + O(kV k)
where l = µQ,0 (d) is a constant defined by (1.15).
Let A(x) = d(x) − µQ,V (d) and B(x) = µQ,V (d) − l. Then d(x) =
ˆ = Â(x) + ˆl(x) + B̂(x). Therefore
A(x) + l + B(x) and, accordingly, d(x)
n
n
n
1 X Â(xj )
1 Xˆ
1 X B̂(xj )
1
t̃(τ ) =
+ 1/3
− 1 + 1/3
l(xj )
M 1/3 j=0 kvj k
M
kvj k
M
kvj k
j=0
j=0
= t̃1 (τ ) + t̃2 (τ ) + t̃3 (τ )
where n = M 2/3 τ . The function A(x)/kv(x)k satisfies the conditions
of Proposition 4.1, in particular µQ,V (A/kvk) = 0, hence t̃1 (τ ) is tight
due to Proposition 4.2 (a). Next
p
1 − 1 − M kVj k2
1
(4.34)
−1=
= O M kVj k2
kvj k
kvj k
To prove the equicontinuity of t̃2 (τ ) we observe that
2/3
t̃2 (τ2 ) − t̃2 (τ1 ) ≤ Const M |τ2 − τ1 | max M kVn k2
1/3
M
n≤TM 2/3
= |τ2 − τ1 | max kṼ (τ )k2 .
τ ≤T
Hence, as before, the function t̃2 (τ ) is Lipschitz continuous with Lipschitz constant max[0,T ] kṼ (τ )k2 that can be bounded by the tightness
of Ṽ (τ ). To prove the equicontinuity of t̃3 (τ ) we use Proposition 4.14
and write
|µQ,V (d) − l| = |µQ,V (d) − µQ,0 (d)| ≤ Const kV k,
hence
t̃3 (τ2 ) − t̃3 (τ1 ) ≤ Const |τ2 − τ1 )| max kṼ (τ )k2
τ ≤T
M 1/3
which is not only bounded by the tightness of Ṽ , but can be made
arbitrarily small.
Proposition 4.2 is proved.
64
N. CHERNOV AND D. DOLGOPYAT
4.6. Second moment. Here we prove Proposition 4.4. We work in
the context of Theorem 2, hence κM = M −δ and n = M 1/2−δ . The
context of Theorem 1 will be discussed in the next section.
Recall that n = ∆M 2/3 . Our first step is to show that under the
conditions of Proposition 4.1
(4.35)
E` Ŝn2 = n DQ̄,V̄ (A) + g ,
where g → 0 as M → ∞ and ∆ → 0 uniformly over all standard pairs
with π1 (γ) ⊂ Υ∗δ2 ,a and length(γ) > M −100 .
Indeed, by (4.31) and Lemma 4.12 (b) we have
n/n
X
E` (D̂Qkn ,Vkn ) + O M 1/3+2δ .
E` Ŝn2 = (1 + o(1)) n
k=0
By Proposition 4.2, for most of the initial conditions
max kQkn − Q̄k, M 2/3 kVkn − V̄ k
k<n/n
is small if ∆ is small, hence for most of the initial conditions x ∈ γ we
have k(x) ≥ n/n thus D̂Qkn ,Vkn (x) = DQkn ,Vkn (A), and so we make a
small error by replacing D̂Qkτ ,Vkτ by DQ̄,V̄ (A) (note that DQ,V (A) is a
bounded and continuous function of Q, V on the domain dist(Q, ∂D) >
r + δ1 ). Thus we obtain (4.35).
Now the parts (a)–(c) of Proposition 4.4 easily follow from Proposition 4.1 (a), (c), and (4.35). To prove (d), we write
!
n−1
X
E` Q̂n − Q̄ = E`
ξj V̂j
j=0
= V̄ E`
n−1
X
j=0
ξj
!
+ E`
n−1
X
j=0
ξj (V̂j − V̄ )
!
= I + II
where ξj = t̂j+1 − t̂j is the intercollision time. To estimate I we use the
notation of the proof of Proposition 4.2 (b) and write:
X E`
ξj = E` ξj − ˆlj /kvj k + E` ˆlj /kvj k = Ia + Ib
As we noted earlier, the function ξj − ˆlj /kvj k satisfies the assumptions
of Proposition 4.1, hence its part (a) implies Ia = O M 1/6+δ . By
using (4.34) and Proposition 4.2 (b) we get
Ib = (1 + o∆→0 (1)) l∆M 2/3 + O M 1/3 .
BROWNIAN BROWNIAN MOTION
65
Next, by the Cauchy-Schwarz inequality and Proposition 4.4 (b)
r X
|II| ≤ Const
E` kV̂j − V̄ k2
j
≤ Const
≤ Const
X √j
j
M
(∆M 2/3 )3/2
≤ Const ∆3/2 .
M
This implies (d). To prove (e), we write, in a similar manner,
hX i X
2 2
2
2
≤ 2 kV̄ k E`
ξj
+ 2 E` ξj (V̂j − V̄ )
E` kQ̄n − Q̄k
= I + II.
Then we have
|I| ≤ Const kV̄ k2 M 4/3 ∆2 = O(∆2 )
and
|II| ≤ Const ∆M
2/3
X
E`
ξj2
2 V̂j − V̄ X j
M2
(∆M 2/3 )2
= O ∆3 .
≤ Const ∆M 2/3
2
M
≤ Const ∆M 2/3
Proposition 4.4 is proven.
4.7. Martingale property. To prove Proposition 4.5 we need to show
that for every m ≥ 1, all bounded and Lipschitz continuous functions
B1 , . . . , Bm on the (Q, V ) space, and all times s1 < s2 < · · · < sm ≤
τ1 < τ2 we have
"m
#
!
Y
E
Bi (Q̂(si ), V̂(si )) [M(τ2 ) − M(τ1 )] = 0.
i=1
where E denotes the expectation and
M(τ2 ) − M(τ1 ) = B(Q̂(τ2 ), V̂(τ2 )) − B(Q̂(τ1 ), V̂(τ1 ))
Z τ2
−
(LB)(Q̂(s), V̂(s)) ds
τ1
66
N. CHERNOV AND D. DOLGOPYAT
In other words, we have to show that
#!
#"
"m
J2
X
Y ζj
→0
βj2 − βj1 − M −2/3
Bi Q̃(si ), Ṽ (si )
(4.36) m
i=1
j=J1
as M → ∞, where
βj = B(Q̂j , M 2/3 V̂j ),
and
J1 = M 2/3 τ1 ,
ζj = LB(Q̂j , M 2/3 V̂j ).
J2 = M 2/3 τ2
(see (4.25) for the definition of m(·) and note that B, LB, and Bi are
bounded and continuous functions). Lemma 4.11 allows us to eliminate
the first factor in (4.36) and reduce it to
!
J
X
(4.37)
m βJ − β0 − M −2/3
ζj → 0 as M → ∞
j=0
where J = M 2/3 (τ2 − τ1 ), and we will denote τ2 − τ1 by τ (note that
even if sm = τ1 , the factor Bm (Q̃(sm ), Ṽ (sm )) will be almost constant
on every standard pair taken at time τ1 , so it can be dropped from
(4.37)).
Next we prove (4.37). Given a small constant ∆ > 0 and a large
constant R > 0, we define Q̂0 , Ṽ 0 similarly to Q̂, V̂ but with an additional stopping rule: in the notation of subsection 4.1, every time
k divides [M 2/3 ∆/n], we “remove from the circulation” all the standard pairs `α,k = (γα,k , ρα,k ) where kV k > M −2/3 R for some point
(Q, V ) ∈ π1 (γα,k ) (that is, we add the curve F −kn (γα,k ) to the set Ik ,
see 4.1). Thus, the set Ik will increase, and k(x) decrease, respectively.
However, by Proposition 4.2 (b) we have, uniformly in ∆,
n
o
0
0
sup mes` (Q̂ , V̂ ) 6= (Q̂, V̂ ) → 0,
`
as R → ∞, M → ∞, where the supremum is taken over all standard
pairs ` = (γ, ρ) with length(γ) > M −100 . Hence it is enough to show
that for all large enough R
!
J
X
(4.38)
lim lim m βJ0 − β00 − M −2/3
ζj0 → 0,
∆→0 M →∞
j=0
where
βj0 = B(Q̂0j , M 2/3 V̂j0 ),
β00 = B(Q̄, M 2/3 V̄ ),
ζj0 = LB(Q̂0j , M 2/3 V̂j0 ),
J = M 2/3 τ.
BROWNIAN BROWNIAN MOTION
67
(note that both expressions in parentheses in Eqs. (4.37) and (4.38) are
uniformly bounded by a constant independent of R, because B has a
compact support). To establish (4.38) it is enough to check that for all
large R and uniformly in k ≤ τ /∆


(k+1)L
X
0
0
− βkL
− M −2/3
ζj0  = o(∆),
(4.39)
lim m β(k+1)L
M →∞
j=kL
where L = M 2/3 ∆. To verify (4.39), we can assume, without loss of
generality, that k = 0. Next we expand the function B into Taylor
series about the point (Q̄, M 2/3 V̄ ):
1
(dV )T BV V dV
2
2
2
3
+O kdQk + M kdV k + M 2/3 kdQkkdV k
βL0 − β00 = h∇Q B, dQi + h∇V B, dV i +
where dQ = Q̂0L − Q̄, dV = M 2/3 (V̂L0 − V̄ ), and BV V = ((∂V2i ,Vj B)).
Applying Proposition 4.4 one can easily derive
E` (βL0
−
β00 )
= M
2/3
2
1 X 2
lhV̄ , ∇Q Bi +
∂V2i ,Vj B + o(∆)
σ̄Q̄ (A)
ij
2 i,j=1
= (LB)(Q̄, M 2/3 V̄ ) ∆ + o(∆)
(4.40)
for each standard pair ` = (γ, ρ) with length(γ) > M −100 (note that
kV̄ k < M −2/3 R due to our modified construction of Q̂0 and V̂ 0 , hence
Proposition 4.4 applies). On the other hand, by Proposition 4.2 (b)
h
i
0
2/3 0
2/3
max E` LB(Q̂j , M V̂j ) − LB(Q̄, M V̄ ) = oM →∞,∆→0 (1).
j≤L
Hence
(4.41)
E`
M
−2/3
L
X
j=0
ζj0
!
= LB(Q̄, V̄ ) ∆ (1 + oM →∞,∆→0(1)).
Now (4.40) and (4.41) imply (4.39). Proposition 4.5 is proved.
Proof of Corollary 4.6. Pick a τ ∈ (0, lT) and denote t = M 2/3 τ .
For every x ∈ Ω choose n so that tn ≤ t < tn+1 . Then
√ Q̃∗ (τ ) = Q̂n + O 1/ M
√ = Q̂[t/l] + Q̂n − Q̂[t/l] + O 1/ M
68
N. CHERNOV AND D. DOLGOPYAT
By Proposition 4.2 (b)
mes` kQ̂n − Q̂[tn /l] k >
max
|n1 −n2 |<M 1/3+δ
kQ̂n1 − Q̂n2 k
→0
as M → ∞. By the tightness of Ṽ (τ )
−1/3+2δ
→ 0.
mes`
max
kQ̂n1 − Q̂n2 k > M
|n1 −n2 |<M 1/3+δ
Combining these estimates gives
mes` sup kQ̃∗ (τ ) − Q̃(τ /l)k > ε → 0
τ
as M → ∞. We also claim that
(4.42)
mes` sup kṼ∗ (τ ) − Ṽ (τ /l)k > ε → 0
τ
but this requires a slightly different argument. The tightness of Ṽ (τ )
means that for any ε > 0 and ε0 > 0 there is ε00 > 0 such that
!
mes`
sup
|n1 −n2 |<M 2/3 ε00
kV̂n1 − V̂n2 k > M −2/3 ε
< ε0 .
Hence, as before,
mes`
sup
mes` sup kṼ∗ (τ ) − Ṽ (τ /l)k > ε
τ
!
|n1 −n2 |<M 1/3+δ
kV̂n1 − V̂n2 k > M −2/3 ε
<
+ o(1) <
ε0 + o(1)
as M → ∞. The arbitrariness of ε0 implies (4.42).
Thus each (Q∗ , V∗ ) can be obtained form the corresponding (Q̂, V̂)
by the time change τ → τ /l.
Remark. In the proof of Corollary 4.6 we used the tightness of t̃(τ ),
but it would be enough if the following function
(4.43)
t̃♦ (τ ) = M −1/3−δ/2 t̂[τ M 2/3 ] − l min{τ M 2/3 , kn}
was tight. We will refer to this observation in Section 6.
BROWNIAN BROWNIAN MOTION
69
5. Fast slow particle
Here we prove Theorem 1, which allows the slow particle (the disk)
to move faster than Theorem 2 does. Our arguments are similar to
those presented in Section 4, in fact they are easier, because we only
need to control
the dynamics during time intervals O M 1/2 , instead
2/3
of O M
.
Recall that for the proof of Theorem 1 we set κM to a small constant
independent of M. Observe that Propositions 4.7 and 4.8, as well as
Lemma 4.12, are applicable in the context of Theorem 1, but in the rest
of Section 4 we assumed kV̄ k ≤ aM −2/3 , which is not the case anymore.
2
Instead of that, we will now assume that M kV̄ k√
≤ 1 − δ2 (and 1 −
δ2 > η).
p where
p We consider the dynamics up to n ≤ T M collisions,
2
T = c 1 − η /l and c is defined in Theoremp1 (note that 1 − η 2 is
the initial speed of the light particle, hence l/ 1 − η 2 will approximate
the mean intercollision time).
The following statement is analogous to Proposition 4.1.
Proposition 5.1. Assume the conditions of Proposition 4.1 but with
2
the modified bound on
√ the initial velocity: M kV̄ k ≤ 1 − δ2 . Then,
uniformly for n ≤ T M , we have
(a) E` Ŝn = O M δ .
(b) E` Ŝn2 = O(n).
(c) E` Ŝn4 = O(n2 ).
Proof. It is enough to divide [0, T] into intervals of length ∆ and apply
Lemma 4.12 to each of them.
Next we define certain continuous functions on the interval [0, T],
in a way similar to (4.5)–(4.6), but with the scaling factors specific to
Theorem 1:
#
"
1/2
l
min{τ
M
,
kn}
p
V0 ,
Q̃(τ ) = M 1/4 Q̂τ M 1/2 − Q0 −
1 − η2
h
i
Ṽ (τ ) = M 3/4 V̂τ M 1/2 − V0 , S̃(τ ) = M −1/4 Ŝτ M 1/2 ,
and
"
#
1/2
l
min{τ
M
,
kn}
p
t̃(τ ) = M −1/4 t̂[τ M 1/2 ] −
1 − η2
The next result is analogous to Proposition 4.2, and the proof only
requires obvious modifications:
70
N. CHERNOV AND D. DOLGOPYAT
Proposition 5.2. (a) For every function A ∈ R satisfying the assumptions of Proposition 4.1, the family of functions S̃(τ ) is tight;
(b) the families Q̃(τ ), Ṽ (τ ), and t̃(τ ) are tight.
The following result is similar to Proposition 4.4:
√
Proposition 5.3. Let ∆ be a small positive constant and n = ∆ M .
The following estimates hold uniformly for all standard pairs ` = (γ, ρ)
with length(γ) > M −100 and π1 (γ) ⊂ Υδ2 , and all (Q̄, V̄ ) ∈ π1 (γ):
1
(a) E` V̂n − V̄ = O
.
M 1−δ
2
−3/2
(b) E` (V̂n − V̄ )(V̂n − V̄ )T = (1+o∆→0 (1)) σ̄Q̄,
.
V̄ (A) ∆ M
(c) E` kV̂n − V̄ k4 = O M −3 ∆2 .
(d) E` Q̂n − Q̄ − t̂n V̄ = O M −1/4 ∆3/2 .
In particular, if V̄ = V0 + uM −3/4 , then
(1 + o
∆→0 (1)) lnu
+ O M −1/4 ∆3/2 .
E` Q̂n − Q̄ − t̂n V0 = p
1 − η 2 M 3/4
(e) E` kQ̂n − Q̄ − t̂n V̄ k2 = O M −1/2 ∆3 .
The proof goes along the same lines as that of Proposition 4.4. We
only note P
that the proofs of parts (d) and (e) do not have to deal with
the term
j ξj V̄ since it is included in t̂n V̄ , whereas the bound on
P
j ξj (Vj − V̄ ) is obtained exactly as before. Also note that the second
estimate of part (d) followspfrom the first one and the fact that, by
Lemma 4.12, E` (t̂n ) ∼ (nl)/ 1 − η 2 .
The next statement is an analogue of Proposition 4.5:
Proposition 5.4. The function Ṽ (τ ) weakly converges, as M → ∞, to
a Gaussian stochastic process Ṽ(τ ) with independent increments, zero
mean, and the covariance matrix
Z τ
2
Cov (Ṽ(τ )) = (1 − η )
σ̄ 2 † “ √ 2 ” (A) ds.
0
Q
sl/
1−η
Proof. Since Ṽ (τ ) is tight, we only need to prove the convergence of
finite dimensional distributions. Fix a τ < T and choose ∆ τ so
that τ /∆ ∈ N. Denote
Rk0 = V̂(k+1)∆√M −M δ − V̂k∆√M ,
BROWNIAN BROWNIAN MOTION
71
and
0
Ṽ (τ ) =
τ /∆
X
Rk0 .
k=0
0
δ−1/4
Note that Ṽ (τ ) − Ṽ (τ ) = O(M
) → 0 as M → ∞, hence the ran0
dom processes Ṽ (τ ) and Ṽ (τ ) must have the same finite dimensional
limit distributions. By the continuity theorem, it is enough to prove
the pointwise convergence of the corresponding characteristic functions,
which we do next.
For every vector z ∈ R2 we write Taylor expansion
(5.1)
M 3/2 hz, Rk0 i2
exp iM 3/4 hz, Rk0 i = 1+iM 3/4 hz, Rk0 i−
+O M 9/4 hz, Rk0 i3 .
2
Lemma 5.5. For any standard pair ` = (γ, ρ) satisfying the conditions
of Proposition 5.3 we have
where
(1 − η 2 ) zT Dk z
E` exp(iM 3/4 hz, Rk0 i) = 1 −
∆ + o(∆).
2
Dk = σ̄ 2 † “
Q
lk∆/
√
1−η 2
” (A)
Proof. We apply Proposition 5.3 (a) and (b) to the linear and quadratic terms of (5.1), respectively, and bound the remainder term by
the Hölder inequality:
3/4
3
E` M 9/4 |hz, Rk0 i| ≤ M 9/4 E` hz, Rk0 i4
and then use Proposition 5.3 (c). A delicate point here is to deal with
2
the matrix σ̄Q̄,
(A) that comes from Proposition 5.3 (b). According to
V̄
Proposition 5.2, for most of the standard pairs ` = (γ, ρ)
p
Q̄ = Q† lk∆/ 1 − η 2 + O M −1/4+δ , V̄ = V0 + O M −3/4+δ
where k∆ is the time moment at which Proposition 5.3 (b) was applied.
2
(A) is a bounded continuous function on the domain {Q, V :
Since σ̄Q̄,
V̄
2
(A) by
dist(Q̄, ∂D) > r + δ2 }, see Appendix A, we can replace σ̄Q̄,
V̄
σ̄ 2 † “
Q
lk∆/
√
”
(A)
1−η 2 ,V0
= (1 − η 2 )Dk
the last equation follows from (1.14).
Now let
Φk (z) = ln E` exp iM 3/4 hz, Rk0 i
,
72
N. CHERNOV AND D. DOLGOPYAT
then by (5.1)
(1 − η 2 ) zT Dk z
∆ + o(∆).
2
Let 0 ≤ τ 0 < τ 00 ≤ T be two moments of time such that k 0 = τ 0 /∆ ∈ N
and k 00 = τ 00 /∆ ∈ N. Then
Φk (z) = −
(5.2)
00
ln E` exp iM
3/4
D
0
00
0
0
z, Ṽ (τ ) − Ṽ (τ )
E
= ln E`
k
Y
exp iM
k=k 0
= (1 + o∆→0 (1))
3/4
hz, Rk0 i
00
k
X
Φk (z),
k=k 0
where we used the same trick as in the proof of Lemma 4.11. By using
(5.2) and letting ∆ → 0 we prove that for any 0 ≤ τ 0 < τ 00 ≤ T
D
E
ln E` exp iM 3/4 z, Ṽ 0 (τ 00 ) − Ṽ 0 (τ 0 )
converges to
1 − η2
−
2
Z
τ 00
τ0
zT σ̄ 2 † “ √
Q
sl/
1−η 2
” (A)z ds.
as M → ∞. This shows that the increments of the limit process are
Gaussian.
Next, let 0 ≤ τ1 < · · · < τm+1 ≤ T be arbitrary time moments
and z1 , . . . , zm ∈ R2 arbitrary vectors. A similar computation as in
Lemma 5.5 shows that the joint characteristic function of several increments
!!
m D
E
X
zj , Ṽ 0 (τj+1 ) − Ṽ 0 (τj )
E` exp iM 3/4
j=1
converges to
!
m Z
1 − η 2 X τj+1 T 2 “
exp −
zj σ̄ † √ 2 ” (A)zj ds .
Q sl/ 1−η
2 j=1 τj
as M → ∞. Hence the increments of the limiting process are independent. This completes the proof of Proposition 5.4.
Lastly, the same argument as in the proof of Corollary 4.6 shows
that the velocity function V(τ )p
defined in Section 1.3 converges to the
stochastic process V(τ ) = Ṽ(τ 1 − η 2 /l). The properties of V listed
in Theorem 1 immediately follow from those
R τ of Ṽ, which we proved
above. The convergence of Q(τ ) to Q(τ ) = 0 V(s) ds follows from the
!
BROWNIAN BROWNIAN MOTION
73
fact that the integration is a continuous map on C[0, lT]. Theorem 1
is proved.
6. Small large particle
Here we prove Theorem 3, which requires the larger particle (the
disk) shrink as M → ∞.
First of all, the results of Section 4 apply to every r ∈ (0, r0 ), where
r0 is a sufficiently small constant, and every time interval (0, T). We
now fix T0 > 0 and for each r ∈ (0, r0 ) apply those results to the time
interval (0, T) with
(6.3)
T = Tr = r−1/3 T0
In other words, we consider a family of systems Fr : Ωr → Ωr
(parametrized by r) to which the results of Section 4 are applied, on
the interval (0, Tr ) with Tr given by (6.3) for each r. Of course, all the
O(·) estimates in Section 4 will now implicitly depend on r.
Next, for each r we define continuous functions on the interval [0, T0 ],
by the following rules that modify (4.5) and (4.43):
(6.4)
Q̃(τ ) = Q̂τ r−1/3 M 2/3 ,
Ṽ (τ ) = r−1/3 M 2/3 V̂τ r−1/3 M 2/3 ,
and
(6.5) t̃♦ (τ ) = r1/6 M −1/3−δ/2 t̂[τ r−1/3 M 2/3 ] − l min{τ r−1/3 M 2/3 , kn}
Now for each r ∈ (0, r0 ) we pick a function Ar ∈ Rr satisfying the
assumptions of Proposition 4.1 with T = r−1/3 T0 and an additional
assumption
(6.6)
sup max{kAk∞ , ωA } < ∞
0<r<r0
where ωA was defined in (4.3). Now we define
(6.7)
S̃(τ ) = r1/6 M −1/3 Ŝτ r−1/3 M 2/3
Proposition 6.1. (a) Given a family of functions A = {Ar } as above,
there is a function MA (r) such that for r < r0 , M > MA (r), the family
S̃(τ ) is tight; (b) There is a function M0 (r) such that for r < r0 ,
M > M0 (r), the families Q̃(τ ), Ṽ (τ ), and t̃♦ (τ ) are tight.
The proof follows the same lines as that of Proposition 4.2, and we
only discuss steps which require nontrivial modifications. The inequality (4.33) now implies (4.32) whenever 2−m < r4/21 M −8/21 , cf. (6.6).
For the case 2−m < r4/21 M −8/21 we need the following sharpened version of Lemma 4.13:
74
N. CHERNOV AND D. DOLGOPYAT
Lemma 6.2. Given a family A = {Ar } as above, there is a function
MA (r) such that for all r < r0 , M > MA (r), all n1 , n2 such that |n2 −
n1 | > r−1/7 M 2/7 and all standard pairs ` = (γ, ρ) with length(γ) >
M −100 and π1 (γ) ⊂ Υ∗δ2 ,a we have
h
i4 ≤ 3ωA2 (n2 − n1 )2 .
E` Ŝn2 − Ŝn1
Proof. This bound follows from Lemma 4.12 (d) and the argument
used in the proof of Proposition 4.1 (d). Note that the term O (n1.9 )
in Lemma 4.12 (d) implicitly depends on r, i.e. it is < C(r) n1.9 , but
we can always increase M0 (r) so that n0.1 > M 0.04 > C(r)/ωA2 , hence
C(r) n1.9 < ωA2 n2 , as desired.
Now the tightness of S̃(τ ) follows due to (6.6).
To prove the tightness of Ṽ (τ ), we need to modify the above argument slightly. Due to (1.8), (4.14) and Lemma 6.2
h
i4 2
E` V̂n2 − V̂n1
≤ ωA
M −4 (n2 − n1 )2 ≤ Const r2 M −4 (n2 − n1 )2 .
where we used (1.26). Therefore,
h
i4 E` Ṽ (τ2 ) − Ṽ (τ1 )
≤ Const (τ2 − τ1 )2 .
which is sufficient to prove the equicontinuity of Ṽ (τ ).
The tightness of Q̃(τ ) and t̃(τ ) follows by the same arguments as the
one in the proof of Proposition 4.2, except we do not have (6.6) for the
function A(x) = d(x) − µQ,V (d), so we cannot obtain the tightness of
t̃1 (τ ). Instead, we use the extra factor M −δ included into the formula
for t̃♦ (τ ) and prove (6.6) for the function A♦ (x) = M −δ/2 A(x). To
this end we observe that ωA♦ = M −δ ωA and choose M0 (r) so that
M0δ (r) > ωA for every r < r0 , hence ωA♦ < 1.
Also, note that Lemma 4.12 (b) still holds, with some g → 0, because we can use the same trick as above – just increase M0 (r) to
suppress the terms depending on r. Next, having proved the tightness and Lemma 4.12 (b), we can derive estimates similar to those of
Proposition 4.4.
The following statement is analogous to Proposition 5.4:
Proposition 6.3. The function Ṽ (τ ) weakly converges, as r → 0 and
M → ∞, M > M0 (r), to the random process σ0 wD (lτ ), in the notation
of (1.28).
BROWNIAN BROWNIAN MOTION
75
The proof is similar to that of Proposition 5.4. A slight complication comes from the fact that, unlike Theorem 1, we have to stop the
heavy particle when it comes too close to the border ∂D. Thus we cannot argue the independence as before, since the increments depend on
whether we have already stopped our particle or not. To overcome this
complication, we let w̄(τ ) be the standard two dimensional Brownian
motion (independent of our dynamical system) and define
(
Vn
if n ≤ kn
Vn♦ =
1/3
−2/3
Vkn + r M
σ0 [w̄(ln) − w̄(lkn)] otherwise
(in other words, rather than terminating the velocity process once the
particle comes too close to the border, we switch to an auxiliary Brownian motion). After this modification, the limiting process will have
independent increments, and we can proceed as in the proof of Proposition 5.4.
Lastly, the same argument as in the proof of Corollary 4.6 (see also
the remark after it) shows that the limit process of the functions V(τ )
defined in Section 1.3 and that of Ṽ (τ ) above only differ by a time
rescaling, τ 7→ τ /l, hence V(τ ) converges to the stochastic process
σ0 wD (τ ), as claimed by Theorem 3. Finally, (1.29) follows by the fact
that the integration is a continuous map on C[0, lT]. Theorem 3 is
proved.
7. Open problems
Here we discuss possible extensions of our results as well as some
open problems.
7.1. Regularity of the diffusion matrix. The obvious question raised
by Theorem 2 is the wellposedness of the equation (1.24). This clearly
2
depends on the regularity of the diffusion matrix σ̄Q
(A). There is experimental evidence (supported by heuristic arguments), see [4], that
this function is not very regular, at least not C 1 , but it is likely to
be Lipschitz continuous. In general, no rigorous results seem to be
available on the regularity of dynamical invariants as functions of the
parameters of the model, in the nonuniformly hyperbolic case. Still we
believe that in our system it can be rigorously proven that
kσ̄Q1 (A) − σ̄Q2 (A)k ≤ Const · kQ1 − Q2 k ln kQ1 − Q2 k
which implies the uniqueness of (1.24).
76
N. CHERNOV AND D. DOLGOPYAT
7.2. Billiards with tangencies and collisions of the massive particle with the wall. Another important problem is to understand
2
the behavior of σ̄Q
(A) as the disk P(Q) approaches the boundary of
D, since this would allow us to extend our results beyond the moment
of the first collision of the heavy disk with the wall. It is natural to
assume that this behavior should be controlled by the billiard dynamics in the domain where the heavy particle just touches ∂D at some
point. This domain is still a dispersing billiard table, but two of its
boundary components will be tangent to each other (will make two
cusps). Therefore, it is important to understand the mixing properties
of dispersing billiards with cusps, which is a long standing open problem in billiard theory. There is a heuristic argument [40] that leads
us to believe that discrete time correlations should decays as O(1/n),
2
hence the diffusion matrix σ̄Q
(A) might be, technically, infinite. The
dynamics in billiard tables with cusps appears to be quite delicate and
requires further investigation.
7.3. Longer time scales. In all the results of our paper, the velocity
v of the light particle does not change significantly during the time
intervals we consider, in fact its fluctuations converge to zero in probability as M → ∞. On the basis of heuristic analysis of Section 1.2,
we expect that v would experience changes of order one after O(M )
collisions with the heavy disk. However, we are currently unable to
treat such long intervals, since the error bounds we have in Proposition
2.2 would accumulate beyond O(1), so we need to improve upon this
proposition in order to proceed further. We note that as the velocity
of the light particle experiences changes of order one, the system starts
approaching its natural equilibrium (its behavior is described by the
invariant ergodic measure).
7.4. Stadia and the piston problem. The question of approach to
a thermal equilibrium was recently considered by several authors for
the piston model [21]. In that model a cubical container is divided into
two compartments by a heavy insulating piston, and these compartments contain ideal gases at different temperatures. If the piston were
infinitely heavy, it would not move and the temperature in each compartment would remain constant. However, if the mass of the piston is
finite the temperatures would change slowly due to the energy and momenta exchanges between the particles and the piston. So far not much
is known about the thermalization time needed for the temperatures
to converge to a common limit value.
There is an obvious analogy between the motion of the piston and
that of the heavy disk in our model. The dynamics of ideal gas particles
BROWNIAN BROWNIAN MOTION
77
in each compartment of in the piston model can be made hyperbolic
by appropriate boundary conditions (say, let the container have a form
of the Bunimovich stadium [6]). Then the methods of our paper could
be used. Let us point out, however, that in our case the fluctuation
about the averaged dynamics are diffusive, while in the piston case
nondiffusive fluctuations may develop as follows. Some particles may
move almost parallel to the piston bouncing back and forth between the
flat walls of the container for a long time. If that happens on one side
of the piston but not the other, the pressure balance will be broken,
and the piston may be forced to move on a macroscopic scale.
7.5. Finitely many particles. The analysis of our paper extends
without changes to systems with several heavy disks and one light
particle. Of course, we need to prevent the disks from colliding with
each other or the boundary of the table by restricting our analysis to
a sufficiently short interval of time. Let us, for example, formulate an
analogue of Theorem 2 in this situation (similar generalizations are possible for Theorem 1 and 3). Let k be the number of heavy disks which
are initially at rest. Then after rescaling time by M 2/3 , the velocity of
the limiting process satisfy


V1
d  . . .  = σQ1 ...Qk dw
Vk
where w is a standard 2k-dimensional Brownian motion. Notice that
even though the heavy disks are not allowed to approach each other,
each one “feels” the presence of the others through the diffusion matrix
σQ1 ...Qk which depends on the positions of all the disks.
In order to extend our results to systems with several light particles,
one needs to generalize Proposition 2.2. Here we have two possibilities.
One is to work with a discrete time dynamics, then the multiparticle
system is a semidispersing billiard in a higher dimensional space. Very
little is known about mixing rates in such systems, see some results in
[16]. Alternatively, we may work directly with a continuous time system, and in this case we get a direct product of 2D billiards. This would
require obtaining the bounds on continuous time correlation functions,
which should be possible in view of recent results [15, 38].
7.6. Growing number of particles. A more realistic model of Brownian Motion consists of one heavy disk and many light particles, whose
number grows with M. It is also quite reasonable to make the size of
the heavy disk decrease as M grows. Let the diameter of the disk be
r = M −α for a small α > 0 and the number of light particles N = M β
78
N. CHERNOV AND D. DOLGOPYAT
for a small β > 0. Since, in view of (1.2), the heavy disk “remembers”
only the last O(M
√ ) collisions,√it is natural to assume that its velocity
will be of order M /M = 1/ M . Hence
it covers √
a distance of order
√
one during a time interval of order M . Let τ = t M . According to
the calculations of Section 6, the expected number of collisions during
this time interval is of order
√
N M r = M 1/2+α−β .
On the other hand, (1.2) tells that O(M ) is a critical number of collisions. Hence the following conjecture seems reasonable:
Conjecture. Suppose that the initial state of each light particle is chosen independently, so that the position and velocity direction are uniformly distributed and the speedR has an initial distribution with smooth
density ρ0 (v). Denote aj (ρ) = |v|j ρ(v) dv. Then the limiting process
Q(τ ) is
(a) straight motion if β = α + 1/2 − ,
(b) the integral of an Ornstein-Uhlenbeck process
(7.1)
dQ = Vdτ
dV = −νVdτ + σdw
where
ν = c1 a1 (ρ0 ),
σ 2 = c2 a3 (ρ0 )
if β = α + 1/2
(c) a Brownian motion if β = α + 1/2 + .
The justification of this conjecture is straightforward. In fact, part
(a) is in direct analogy with Theorem 1. There are too few collisions
to produce significant changes of the velocity of the massive disk. Part
(b) is similar to Theorem 2, with one notable difference: in Theorem 2,
there is no drift for the velocity of the disk since the number of collisions
−1
was too small for the factor M
in (1.2) to take effect. Under the
M +1
setting of the above conjecture, it is this factor that determines the
drift of the Ornstein-Uhlenbeck process. The factor a1 in the drift
term comes from the fact that the number of collisions of the massive
disk with any given particle is proportional to the speed of that particle.
The reason for the factor a3 in the diffusion term is explained before
Theorem 1 (see also [25]). Also, observe that in the two particle model
treated in this paper, the velocity of the massive disk has a maximal
value, √1M , hence when it gets close to this value it is more likely to
decrease than to increase. In this sense, we have a “superdrift” in
the two particle model. Finally, in the case (c) the Ornstein-Uhlenbeck
regime should take effect on time intervals which are much shorter than
BROWNIAN BROWNIAN MOTION
79
τ , hence (c) is quite natural in view of the fact that Ornstein-Uhlenbeck
process satisfies the central limit theorem.
We believe that the cases of several light particles of the previous
subsection and a growing number of particles discussed here are similar, on a technical level. Indeed, our arguments are based on the
estimation of the first four moments. For arbitrary many particles, the
computation of the fourth moment contains only the contribution of
all 4-tuples of collisions, but each 4-tuple involves at most four different light particles, hence an extension of Proposition 2.2 to only four
light particles should be enough for the study of systems with arbitrary many particles. In the case of a growing number of particles,
there is also an additional complication because there are, inevitably,
slow particles for which there is not enough time for mixing to take
effect. However, we expect the contribution of those particles be small,
since their collisions with the heavy disk will result in relatively small
changes of the velocity of the latter (cf. also [25]).
7.7. Particles of positive size. The results of our paper obviously
remain valid if the light particle has a positive diameter, 2r0 , which is
smaller than the shortest distance between scatterers Bi . Indeed, that
model can be reduced to ours by enlarging the massive disk and all the
scatters by r0 . However, a model of several light particles of positive
diameter becomes more interesting, since the particles can interact with
each other. To fix our ideas, consider the situation of the previous
subsection with β = α + 1/2 but now let us assume that instead of
r0 = 0 we√have r0 = M −γ . Then each light particle is expected to collide
with N r0 M other particles. Since momentum transferred during
√ each
collision is of order 1, now an interesting scaling regime is N r0 M ∼ 1.
In this case we can expect ρ to change according to the kinetic theory.
Thus the following statement seems reasonable.
Conjecture. The limiting process satisfies
(7.2)
dQ = Vdτ
where
(a) ν = c1 a1 (ρ0 ),
dV = −ν(τ )Vdτ + σ(τ )dw
σ 2 = c2 a3 (ρ0 )
if γ > α + 1,
(b) ν = c1 a1 (ρτ ),
σ 2 = c2 a3 (ρτ )
and ρt satisfies the homogeneous Boltzmann equation
dρτ
= Q(ρτ , ρτ ),
dτ
80
N. CHERNOV AND D. DOLGOPYAT
where Q is the Boltzmann collision kernel if γ = α + 1,
(c) ν = c1 a1 (ρMax (a2 (ρ0 ))),
σ 2 = c2 a3 (ρMax (a2 (ρ0 )))
where ρMax (a2 (ρ0 )) is the Maxwellian distribution with the same second
moment as ρ0 if γ < α + 1.
Appendix A. Background on dispersing billiards.
In our paper we have made an extensive use of statistical properties
of dispersing billiards obtained recently in [9, 16, 57]. On several occasions, though, those results were insufficient for our purposes, and
we needed to extend or sharpen them. Here we adjust the arguments
of [9, 16, 57] to obtain the results we need. The reader is advised to
consult those papers for relevant details.
A.1. Decay of correlations. To fix our notation, let D = T2 \ ∪ri=0 Bi
be a dispersing billiard table, where B0 , B1 , . . . , Br are open convex
scatterers with C 3 smooth boundaries and disjoint closures (the scatterer B0 will play a special role, it corresponds to the disk P(Q) in our
main model). Denote by ΩD = ∂D × [−π/2, π/2] the collision space,
FD : ΩD → ΩD the collision map, and µD the corresponding invariant
measure.
Assume that the horizon is finite, i.e. the free path between collisions
is bounded by Lmax < ∞. In this case for every k ≥ 1 the map FDk
is discontinuous on a set Sk ⊂ ΩD , which is a finite union of smooth
compact curves. The complement ΩD \ Sk is a finite union of open
domains which we denote by ΩD,k,j , 1 ≤ j ≤ Jk .
Now let Hk,η denote the space of functions on ΩD which are Hölder
continuous with exponent η on each domain ΩD,k,j , 1 ≤ j ≤ Jk :
f ∈ Hk,η ⇔ ∀j ∈ [1, Jk ] ∀x, y ∈ ΩD,k,j |f (x) − f (y)| ≤ Kf [dist(x, y)]η
One of the central results in the theory of dispersing billiards is an
exponential decay of correlations:
Theorem [57]. For every η ∈ (0, 1] and k ≥ 1 there is a θk,η ∈ (0, 1)
such that for all f, g ∈ Hk,η and n ∈ Z
(A.1)
where
(A.2)
|n|
|µD (f · (g ◦ FDn )) − µD (f )µD (g)| ≤ Cf,g θk,η
Cf,g = (Kf + kf k∞ ) (Kg + kgk∞ )
The exponential bound (A.1) was stated and proved in [16, 57], but
the formula (A.2), which we will need for or purposes, was not explicitly
BROWNIAN BROWNIAN MOTION
81
derived there. It follows from the estimates on pages 608–609 of [57],
and we refer the reader to that paper for details.
Next, we will extend the above result in several ways:
Extension 1. Suppose one scatterer (specifically, B0 ) is removed from
the construction of ΩD , i.e. we redefine Ω̃D = ∪ri=1 ∂Bi × [−π/2, π/2],
and respectively the return map F̃D : Ω̃D → Ω̃D and the invariant
measure µ̃D . Note that we do not change the dynamics – the billiard
particle still collides with the scatterer B0 , we simply skip those collisions in the construction of the collision map. Assume, additionally,
that the billiard particle cannot experience two successive collisions
with B0 without colliding with some other scatterer(s) in between (this
follows from our “finite horizon” assumption in Section 1.3). In this
case the analysis done in [57] goes through and the above theorem now
holds for the dynamical system (Ω̃D , F̃D , µ̃D ) and functions f, g defined
on Ω̃D .
The value of θκ,η in (A.1) depends the following quantities characterizing the given billiard table:
(a) the minimal and maximal free path (called Lmin and Lmax ),
(b) the minimal and maximal curvature of the boundary of the
scatterers,
(c) the upper bound on the derivative of the curvature of the scatterers,
(d) the complexity Kn of the singularity set Sn , which is the maximal number of pieces into which Sn can partition arbitrary
short unstable curves.
It is known [8, 57] that Kn ≤ C1 n + C2 , where C1 and C2 are constants
determined by the number of possible tangencies between successive
collisions, i.e. the by maximal number of points at which a straight line
segment I ⊂ D can touch some scatterers Bi . We note that this number
does not exceed Lmax /Lmin , thus C1 and C2 are effectively determined
by Lmax and Lmin .
Extension 2. Consider a family of dispersing billiard tables obtained
by changing the position of one of the scatterers (specifically, B0 ) continuously on the original dispersing billiard table. We only allow such
changes that the maximal free path Lmax remains bounded away from
infinity, and the minimal free path Lmin remains bounded away from
zero. Then all the characteristic values (a)–(d) of the billiard tables in
our family will effectively remain unchanged, and therefore the bound
(A.1) will be uniform. (Note that the space ΩD does not depend on the
82
N. CHERNOV AND D. DOLGOPYAT
position of the movable scatterer B0 , hence the functions f, g in (A.1)
do not have to change with the position of B0 ).
Extension 3. Suppose we not only change the position of the scatterer
B0 , but also reduce its size homotetically (for simplicity, suppose B0
is a disk of radius r0 , and we replace it with a disk of radius r < r0 ).
Hence we consider a larger family of dispersing billiard tables than in
Extension 2. Now the collision space ΩD depends on the size of B0 , but
we restrict the analysis to the space Ω̃D constructed exactly as we did
in Extension 1, by skipping collisions with B0 . Then the space Ω̃D will
be the same for all billiard tables in our family, so we can speak about
the uniformity of the exponential bound on correlations for the map
F̃. Again, we assume a uniform upper bound on Lmax and a uniform
positive lower bound on Lmin . There are several new problems now:
The curvature of ∂B0 will not be uniformly bounded anymore, it will
be proportional to 1/r. The upper bound on the curvature is used to
prove a uniform transversality of stable and unstable cones, see pp.
534–535 in [16]. This may only cause problems on the boundary of the
scatterer B0 itself, but it was specifically excluded from the construction
of Ω̃D . The upper bound on the curvature is also used in the distortion
and curvature estimates, see Appendix B, but we will show that those
estimates remain uniform over all r > 0, see a remark after the proof
of Lemma B.1. Next, the curvature of the disk B0 is constant, so its
derivative is zero.
Lastly, the complexity Kn of the singularity set Sn will be affected
by r, too, if r is allowed to be arbitrarily small. Indeed, for scatterers
of fixed size, one considers [16, 57] short enough unstable curves that
can only break into two pieces at any collision (one piece collides, the
other passes by, as it is explained in the proof of Lemma 3.10). But
now, no matter how small an unstable curve is, the scatterer B0 may
be even smaller, and then the unstable curve may be torn by B0 into
three pieces. The middle piece hits B0 , gets reflected, and by the next
collision its image will be of length O(1). Thus, the sequence Kn will
grow exponentially fast, and therefore the complexity bound (3.19) may
easily fail.
To get around this problem, we follow the same strategy as in the
proof of the growth lemma 3.10. All we need is to prove the one-step
expansion estimate, the analogue of (3.20):
(A.3)
(n)
∃n ≥ 1 : α0 := lim inf
δ→0
sup
W :length(W )<δ
X
i
θi,n < 1
BROWNIAN BROWNIAN MOTION
83
−1
Here W ⊂ Ω̃D denotes an H-curve and θi,n
the smallest local factor of
−n
n
expansion of F̃D (Wi,n ) under the map F̃D , where Wi,n , i ≥ 1, denote
the H-components of F̃Dn (W ).
First we consider the case n = 1. Collisions of W with the fixed scatterers Bj , j ≥ 1, are described in the proof of Lemma 3.10. Now if W
collides with the disk B0 of a very small radius r, say r < O(length(W )),
then W may be torn into three pieces as described above. The middle piece (reflecting off B0 ) will be further subdivided into countably
many H-components lying in all the homogeneity strips I±k for k ≥ k0 ,
as well as I0 . Those H-components will be expanded by factors ≥
ck 2 /r and > c/r, respectively, see our estimates in Section
P 3.1. Hence
the contribution
of all these H-components to the sum
θi,1 will be
P
r/c + 2r k≥k0 (ck 2 )−1 ≤ Const r.
Thus, the image F̃D (W ) may consist, generally, of the following
H-components Wi,1 : countably many Wi,1 ’s produced by a collision
with B0 , at most Lmax /Lmin countable sets of Wi,1 ’s produced by almost tangential reflections off some fixed scatterers (cf. the proof of
Lemma 3.10), and at most two H-components that miss the collision
with B0 and all the grazing collisions – these land somewhere else on
∂D. The last two H-components are only guaranteed to expand by a
moderate factor of θ −1 , which gives an estimate
(A.4)
(1)
α0 ≤ 2θ +
Lmax Const
+ Const r
Lmin k0
Actually, the third term only comes up if r < O(length(W )) = O(δ),
hence we can take lim supδ→0 and simplify the previous bound as
(1)
α0 ≤ 2θ + Const/k0
The last term can be made arbitrarily small, as in the proof of Lemma 3.10,
but the first term may already exceed one, hence the estimate (A.3)
would fail for n = 1.
Therefore, we have to consider the case n ≥ 2. Our previous analysis
shows that the image F̃Dn (W ) will consist of H-components Wi,n of
two general types: (a) countably many H-components that have either
collided with B0 at least once or got reflected almost tangentially off
some fixed scatterer at least once, and (b) all the other H-components.
Respectively, we decompose Σi θi,n = Σ(a) + Σ(b) .
First, we estimate Σ(a) . The above estimate (A.4) can be easily
extended to a more general bound:
X
D := sup
θi,1 < Const
W
i
84
N. CHERNOV AND D. DOLGOPYAT
where the supremum is taken over all the H-curves W ⊂ Ω̃D . Now the
chain rule and the induction on n gives
Σ(a) ≤ Const Dn (r + 1/k0 )
(A.5)
We now turn to Σ(b) . First, we need to estimate the maximal number
of H-components of type (b), we call it K̃n . Suppose for a moment
that B0 is removed from the billiard table. Then any short u-curve
will be cut into at most Kn0 ≤ C1 n + C2 pieces by the singularities of
the corresponding collision map during the first n collisions, see above,
where C1 and C2 only depend on the fixed scatterers Bi , i ≥ 1. Now we
put B0 back on the table. As we have seen, each unstable curve during
a free flight between successive collisions with the fixed scatterers can
be cut by B0 into three pieces, of which only two (the middle one
excluded) can produce H-components of type (b), thus adding one more
piece to our count. Therefore the total number of pieces of type (b),
after n reflections, will not exceed K̃n ≤ nKn0 ≤ C1 n2 + C2 n. This
gives a quadratic bound on K̃n , and it is important that this bound is
independent of the location or the size of the variable scatterer B0 , i.e.
our bound is uniform over all the billiard tables in our family.
Now, since θi,n ≤ θ n for every H-component of type (b), then
Σ(b) ≤ K̃n θ n ≤ (C1 n2 + C2 n) θ n
thus
(n)
α0 ≤ (C1 n2 + C2 n) θ n + Const Dn /k0
where the lim supδ→0 is already taken to eliminate r from (A.5). Clearly
the first term here is less than one for some n ≥ 1, and then the second
term can be made arbitrarily small by choosing k0 large, hence we
obtain (A.3).
This proves that the exponential bound on correlations for the map
F̃D will be uniform for all the billiard tables in the family constructed
in Extension 3.
We need to make yet another remark: the one-step expansion estimate (A.3) implies the analogue of the growth lemma 3.10 for the map
F̃, with all the constants β1 , . . . , β6 and q independent of the location
or the size of B0 .
A.2. Diffusion matrix. Here we discuss the properties of the matrix
2
σ̄Q
(A) defined by the Green-Kubo formula (1.13).
2
(A) depends on Q continuously.
Lemma A.1. The matrix σ̄Q
BROWNIAN BROWNIAN MOTION
85
Proof. Every term in the series (1.13) depends on Q continuously, and
the claim now follows from a uniform bound proved in Extension 1 of
Section A.1.
2
Next we describe the conditions under which the matrix σ̄Q
(A) is
2
singular. If σ̄Q (A) is singular, it has an eigenvector u with a zero
eigenvalue, and then the function g(x) = hA(x), ui is a coboundary
[9, 7]. On the other hand, it was shown in Section 7 of [9] and Section 5 of [11] that if there is a periodic point x ∈ ΩQ of period k
Pk−1
such that Sk (x) := i=0
g(FQi x) 6= 0 and g is smooth at the points
x, . . . , FQk−1 (x), then g cannot be a coboundary. Therefore, we obtain
the following:
2
Criterion for nonsingularity of σ̄Q
(A). Suppose there are two periodic points, x1 ,P
x2 ∈ Ω with periods k1 and
, respectively, such that
Pkk22 −1
k1 −1
i
the vectors u1 = i=0 A(FQ x1 ) and u2 = i=0 A(FQi x2 ) are nonzero
2
and noncollinear. Then σ̄Q
(A) is nonsingular.
Observe that there is at least one periodic point satisfying this condition, it is made by an orbit running between P(Q) and the closest
scatterer Bi . On many billiard tables, one can find several such trajec2
tories, which would guarantee the nondegeneracy of σ̄Q
(A).
2
Next we discuss the asymptotics of σQ (A) as r → 0 and prove (1.26),
in fact a stronger version of it:
8r
2
σQ
(A) =
I + O(r2 | ln r|),
3 Area(D)
which, by virtue of (1.15), is equivalent to
8πr
2
(A.6)
σ̄Q
(A) =
I + O(r2 | ln r|).
3 length(∂D)
First we fix our notation. To emphasize the dependence of our dynamics on r we denote by ΩQ,r the collision space, FQ,r the collision map
and µQ,r the invariant measure. We also use notation of Extension 3
of Section A.1, after identifying our disk P(Q) with the variable scatterer B0 : thus we get the collision space Ω̃D , the collision map F̃D on it
(however, we will denote this map by F̃Q,r to emphasize its dependence
on Q and r), and the corresponding invariant measure µ̃D . Note that
µ̃D is obtained by conditioning the measure µQ,r on Ω̃D , the ratio of
their densities is
length(∂D) + 2πr
= 1 + O(r)
Lr :=
length(∂D)
and µ̃D is in fact independent of Q and r.
86
N. CHERNOV AND D. DOLGOPYAT
Consider the function Ã(x) : = A(FQ(x)) on Ω̃D and the matrix
∞ Z
T
X
2
n
(A.7)
σ̃Q (Ã) :=
à à ◦ F̃Q,r
dµ̃D .
n=−∞
Ω̃D
2
2
It follows from Theorem 1.3 in [41] that σ̃Q
(Ã) = Lr σ̄Q
(A). Hence
it is enough to prove that
8πr
2
(A.8)
σ̃Q
(Ã) =
I + O(r2 | ln r|).
3 length(∂D)
Due to the invariance of the measure µQ,r under the map FQ,r , we
have µ̃D (Ã) = µQ,r(A) = 0. It is easy to check that à is Hölder
continuous with exponent η = 1/2 and coefficient KÃ = Const/r, hence
à ∈ H1,1/2 , in the notation of Section A.1. Therefore, the uniform
bound on correlations proved in Extension 3 gives
Z
T
|n|
n
dµ̃D ≤ Const r−2 θ1,1/2
à à ◦ F̃Q,r
Ω̃D
K| ln r|
where Const is independent of Q and r. Let K be such that θ1,1/2 = r4 .
Then
T
X Z
2
n
σ̃Q (Ã) =
dµ̃D + O(r2 ).
à à ◦ F̃Q,r
|n|≤K| ln r|
Ω̃D
Next we prove that for each n 6= 0
Z
T
n
(A.9)
dµ̃D ≤ Const r2
à à ◦ F̃Q,r
Ω̃D
By the time symmetry it is enough to consider n > 0. The domain
{x : Ã 6= 0} can be foliated by H-curves {γ} of length O(r) which will
land on ∂P (Q) under the map FQ,r . If the foliation is smooth enough,
the conditional measures {mesγ } will have homogeneous densities, cf.
Section 3.3.
n
(x) to
Let γ be one of those H-curves and rn (x) the distance from FQ,r
n
n
the nearest endpoint of the H-component of FQ,r(γ) containing FQ,r(x),
cf. Section 3.4. The first image FQ,r (γ) consists of countably many H−1
components γk ⊂ Ik for k = 0 and |k| ≥ k0 . Recall that the map FQ,r
contracts the curve γ0 by a factor < c/r and the curves γk by a factor
< ck 2 /r for some c > 0. Therefore,
!
∞
X
ε
mesγ (x : r1 (x) < ε) < Const ε + 2
≤ Const ε
2
k
k=k
0
Lemma 3.10 then implies mesγ (x : r2 (x) < ε) < Const ε.
BROWNIAN BROWNIAN MOTION
87
n
Now, let r̃n (x) be the distance from F̃Q,r
(x) to the nearest endpoint
n
n
of the H-component of F̃Q,r (γ) containing F̃Q,r
(x). Note that r̃1 (x) =
r2 (x), hence mesγ (x : r̃1 (x) < ε) < Const ε. According to the analogue
of the growth lemma 3.10 for the map F̃ , which was mentioned in the
end of Section A.1, we have
mesγ (x : r̃n (x) < ε) ≤ Const ε
for all n ≥ 1. Now since the disk P(Q) has size O(r), the fraction of
the full image of γ that collides with P(Q) at any given time n ≥ 1
will be O(r), with respect to the measure mesγ . Therefore,
∗
n
µ̃D x ∈ Ω̃ : Ã ◦ F̃Q,r (x) 6= 0 ≤ Const r µ̃D (Ω̃∗ ) ≤ Const r2
This implies (A.9).
It remains to compute the n = 0 term
Z
Z
T
à à dµ̃D = Lr
A AT dµQ,r
(A.10)
Ω̃D
Ω∗Q,r
where Ω∗Q,r = ∂P (Q) × [−π/2, π/2] is the collision space of the disk
P(Q). The measure µQ,r has density c−1 cos ϕ dr dϕ in the coordinates
r, ϕ introduced in Section 3.2, where c = 2 length(∂D) + 4πr is the normalization factor. For convenience, we replace the arclength parameter
r on ∂P(Q) with the angular coordinate ψ ∈ [0, 2π), then we get the
ψ, ϕ coordinates on Ω∗Q,r and dµQ,r = c−1 r cos ϕ dψ dϕ.
Due to an obvious rotational symmetry, the matrix (A.10) is a scalar
multiple of the identity matrix, so it is enough to compute its first diagonal entry. The first component of the vector function A is 2 cos ψ cos ϕ,
hence the first diagonal entry of (A.10) is
Z π/2 Z 2π
r
8πr
4 cos2 ψ cos3 ϕ dψ dϕ =
2 length(∂D) −π/2 0
3 length(∂D)
This completes the proof of (A.8), and hence (1.26).
A.3. The function space R. Here we prove Lemma 2.1. Clearly,
it is enough to prove it for B1 ≡ 1. We use induction on n0 . For
n0 = 1 the lemma reduces to the definition of R. For n0 ≥ 2 we put
B = B2 ◦ F n0 −2 and A = B2 ◦ F n0 −1 . The Hölder continuity of A on
the connected components of Ω \ Sn0 follows from (2.7):
|A(x) − A(x0 )| = |B(F (x)) − B(F (x0 ))|
≤ KB [dist(F (x), F (x0 ))]αB
≤ KB KFαB [dist(x, x0 )]αB αF
88
N. CHERNOV AND D. DOLGOPYAT
It remains to estimate the local Lipschitz constant Lipx (A) defined by
(2.8). First we note that Lipx (A) ≤ kDx F k Lipy (B), where y = F (x).
The derivative Dx F is unbounded in the vicinity of S, more precisely,
on one side of S which corresponds to nearly grazing collisions, i.e.
where y is close to ∂Ω. Denote d1 = dist(x, S), d2 = dist(x, Sn0 \ S),
d3 = dist(y, ∂Ω) ∼ π/2 − |ϕ|, where y = (r, ϕ) in the notation of
Section 3.2, and d4 = dist(y, Sn0 −1 ). All these distances are measured along some unstable√curves, see Section 3.5, and Dx F attains its
maximal
√ expansion ∼ 1/ d1 along unstable curves through x, hence
d3 ∼ d1 . Note that d := dist(x, Sn0 ) = min{d1 , d2 }. We now have
two cases:
(a) If d1 < d2 , then d3 < Const d4 , hence dist(y, Sn0 −1 ) > Const−1 d3
and
Const Const
Const
Lipx (A) ≤ √
≤ (1+β )/2
βB
B
d
d1 d3
√
(b) If d2 ≤ d1 , then d4 ≤ Const d3 and d4 ∼ d2 / d1 , hence
Const
Const Const
Const
Lipx (A) ≤ √
≤ 1/2−β /2 β /2 ≤ (1+β )/2
βB
B
B
d
d1 d4
d1
d2 B
In either case we obtain the required estimate with βA = (1 + βB )/2.
Since βB < 1, we also have βA < 1. Lemma 2.1 is proved.
Lastly, we prove Proposition 4.14. Its first claim follows from the fact
that the configuration space of our system is a four dimensional domain
bounded by cylindrical surfaces. To prove the Lipschitz continuity
of µQ,V (d), consider two nearby points (Q, V ) and (Q0 , V 0 ) and set
h = kQ − Q0 k + kV − V 0 k. First assume that the light particle starts
at a point (q, v) such that q ∈ ∂D and compare d(Q, V, q, v) with
d(Q0 , V 0 , q, v). It is convenient to use the coordinate frame moving with
velocity vector V (in this frame, the disk Q, V is at rest). Then the
light particle moves with velocity v − V and the disk Q0 , V 0 moves with
velocity V 0 −V . At the time of the next collision, the moving disk Q0 , V 0
will be at distance O(h) from the fixed disk Q, V . One can check now by
direct inspection that the average difference d(Q, V, q, v)−d(Q0 , V 0 , q, v)
is O(h). The other case q ∈ ∂P(Q) is even easier.
Appendix B. Distortion bounds.
Here we prove rather technical Propositions 3.6, 3.7 and Lemma 3.20
whose proofs were left out in Section 3.
Let x0 = (Q0 , V0 , q0 , v0 ) ∈ Ω and x00 = (Q00 , V00 , q00 , v00 ) ∈ Ω be two
nearby points that belong in one homogeneity section Im0 . Assume
that for each 1 ≤ i ≤ n the points xi = (Qi , Vi , qi , vi ) = F i (x0 ) and
BROWNIAN BROWNIAN MOTION
89
x0i = (Q0i , Vi0 , qi0 , vi0 ) = F i (x00 ) also belong in one homogeneity section,
call it Imi . We assume that for 0 ≤ i ≤ n
kQi − Q0i k ≤ C kqi − qi0 k/M
and
kVi − Vi0 k ≤ C kvi − vi0 k/M
where C > 1 is a large constant. We denote by (ri , ϕi ) and (ri0 , ϕ0i ) the
coordinates of the points π0 (xi ) and π0 (x0i ), respectively, and put
p
δi = (ri − ri0 )2 + (ϕi − ϕ0i )2
We assume that for all i ≤ n
(B.1)
kqi − qi0 k ≤ Cδi
and
(B.2)
kvi − vi0 k ≤ Cδi
where C > 1 is a large constant. These assumptions hold, for example,
when xi and x0i belong in one standard unstable curve (this follows from
Propositions 3.1 and 3.4).
Let dx0 = (dQ0 , dV0 , dq0 , dv0 ) be a postcollisional standard unstable tangent vector at x0 , and dx00 = (dQ00 , dV00 , dq00 , dv00 ) a similar vector at x00 . For i ≥ 1, denote by dxi = (dQi , dVi , dqi , dvi ) and dx0i =
(dQ0i , dVi0 , dqi0 , dvi0 ) their postcollisional images at the points xi and x0i ,
respectively. We say that the standard unstable vectors dxi and dx0i
are (εi , ε̃i )-close, if the following four bounds hold:
kdqi − dqi0 k ≤ εi kdqi k
kdvi − dvi0 k ≤ εi kdvi k
kdQi − dQ0i k ≤ ε̃i kdqi k/M
kdVi − dVi0 k ≤ ε̃i kdvi k/M
Lemma B.1 (One-step distortion control). Assume that the standard
unstable vectors dx0 and dx00 are (ε0 , ε̃0 )-close for some small ε0 , ε̃0 > 0.
Then their images dx1 and dx01 will be (ε1 , ε̃1 )-close, where
2ε̃0
C
δ0 + δ 1
(B.3)
ε 1 = ε0 +
1+ √
+C
M
cos ϕ1
M
C
δ0 + δ 1
(B.4)
ε̃1 = (2ε0 + ε̃0 ) 1 + √
+C
cos ϕ1
M
Here C > 0 is a large constant.
90
N. CHERNOV AND D. DOLGOPYAT
−
−
−
−
Proof. We first compare the precollisional vectors dx−
1 = (dQ1 , dV1 , dq1 , dv1 )
0 −
0 −
0 −
0 −
0 −
0
and (dx1 ) = ((dQ1 ) , (dV1 ) , (dq1 ) , (dv1 ) ) at the points x1 and x1 ,
respectively. Equation (3.2) and the triangle inequality imply
kdq1− − (dq10 )− k ≤ ε0 kdq0 k + s ε0 kdv0 k
+|s − s0 | kdv00 k
where s (resp., s0 ) is the time between collisions at the points x0 and
x1 (resp., x00 and x01 ). Due to Proposition 3.1 (d)–(e), the vectors dq0
and dv0 are almost parallel, for large M , hence we can combine the
first two terms in the above bound:
√
ε0 kdq0 k + s ε0 kdv0 k ≤ ε0 (1 + C/ M ) kdq1− k
Here and below we denote by C = C(∂D, r) > 0 various constants.
Next, it is a simple geometric fact that
ksv − s0 v 0 k ≤ δ0 + δ1 + kQ0 − Q00 k + kQ1 − Q01 k
and by the assumptions (B.1)–(B.2) we get
|s − s0 | ≤ C(δ0 + δ1 )
Using Proposition 1 (g) gives
(B.5)
√
kdq1− − (dq10 )− k ≤ ε0 (1 + C/ M ) kdq1− k + C(δ0 + δ1 ) kdq1−k
Similarly,
(B.6)
√
0 −
−
−
kdQ−
1 − (dQ1 ) k ≤ ε̃0 (1 + C/ M ) kdq1 k/M + C(δ0 + δ1 ) kdq1 k/M
where the estimation of the last term involves Proposition 3.1 (c).
The postcollisional vectors dq1 and dQ1 depend on the precollisional
vectors dq1− and dQ−
1 through certain reflection operators defined in
terms of the normal vector n, see Section 3.1. Since ∂D and ∂P(Q)
are C 3 smooth, those reflection operators depend smoothly on x1 , with
bounded derivatives, hence
0 −
kdq1 − dq10 k ≤ kdq1− − (dq10 )− k + 2kdQ−
1 − (dQ1 ) k
+Cδ1 kdq1− k + Cδ1 kdQ−
1k
Applying (B.5)–(B.6) and Proposion 3.1 (b) gives
√
kdq1 − dq10 k ≤ (ε0 + 2ε̃0 /M )(1 + C/ M ) kdq1 k
+C(δ0 + δ1 ) kdq1 k
Similarly,
0 −
−
0 −
kdQ1 − dQ01 k ≤ kdQ−
1 − (dQ1 ) k + 2kdq1 − (dq1 ) k/M
−
+Cδ1 kdQ−
1 k + Cδ1 kdq1 k/M
BROWNIAN BROWNIAN MOTION
91
Applying (B.5)–(B.6) and Proposion 3.1 (b) gives
√
kdQ1 − dQ01 k ≤ (2ε0 + ε̃0 ) (1 + C/ M ) kdq1 k/M
+C(δ0 + δ1 ) kdq1 k/M
We now consider the velocity components dv and dV . They do
not change between collisions. At collisions, these vectors are mapped
by certain reflection operators defined in terms of the normal n and
acquire an addition involving the operator Θ− , see Section 3.1. In
those equations all the operators and vectors smoothly change with
the point x1 with bounded derivatives (see also (B.2)), except for the
unbounded factor kw + k/hw + , ni, which we later denoted by 1/ cos ϕ.
In what follows, we apply an elementary estimate
1
1 |ϕ1 − ϕ01 |
Cδ1
−
(B.7)
≤
cos ϕ1 cos ϕ0 cos ϕ1 cos ϕ0 ≤ cos2 ϕ1
1
1
Then, in the (simpler) case of a collision of the light particle with ∂D,
we obtain
kdv1 − dv10 k ≤ kdv0 − dv00 k + kΘ− k kdq1− − (dq10 )− k
(B.8)
+C δ1 kdv1 k + CkΘ− k kdq1−k δ1 / cos ϕ1
where Θ− is the operator (3.3) at the point x1 . By using (B.5), the
first two terms on the right hand side in the above inequality can be
bounded as follows:
A : = kdv0 − dv00 k + kΘ− k kdq1− − (dq10 )− k
√
≤ ε0 kdv0 k + ε0 (1 + C/ M ) kΘ− k kdq1− k + C(δ0 + δ1 ) kΘ− k kdq1− k
It is clear from (3.3) that
kΘ− k =
2κkv + k2
hv + , ni
where all the vectors are taken at the point x1 . It is also clear that the
operator Θ− attains its norm on the vectors perpendicular to v − = v1− .
By Proposition 3.1, the vector dq1− is almost perpendicular
to v1− . Thus
√
kΘ− k kdq1− k = (1 + γ)kΘ− (dq1− )k with some γ = O(1/ M ). Now we
can combine the first two terms on the right hand side of the previous
inequality as follows:
√
A0 : = ε0 kdv0 k + ε0 (1 + C/ M ) kΘ− k kdq1− k
√
≤ ε0 kRn (dv0 )k + ε0 (1 + C/ M ) kΘ+ (dq1+ )k
√
≤ ε0 (1 + C/ M ) kdv1 k
92
N. CHERNOV AND D. DOLGOPYAT
Finally, combining all our estimates gives
√
(B.9) kdv1 − dv10 k ≤ ε0 (1 + C/ M ) kdv1 k + C(δ0 + δ1 ) kdv1 k/ cos ϕ1
In the (more difficult) case of an interparticle collision we have a few
extra terms in the main bound (B.8):
kdv1 − dv10 k ≤ · · · + 2kdV0 − dV00 k + Cδ1 kdV0 k
−
−
0 −
+Cδ1 kΘ− k kdQ−
1 k + kΘ k kdQ1 − (dQ1 ) k
where · · · denote the terms already shown in (B.8). Now, the first new
term above is bounded by
(B.10)
2kdV0 − dV00 k ≤ 2ε̃0 kdv0 k/M
The following two terms can be easily bounded and incorporated into
0 −
the previous estimate (B.9). The last term kΘ− k kdQ−
1 − (dQ1 ) k is
bounded, with the help of (B.6), by
√
ε̃0 (1 + C/ M ) kΘ− k kdq1− k/M + C(δ0 + δ1 ) kΘ− k kdq1 − k/M
The second term in this expression can be easily incorporated into the
previous estimate (B.9). To the first term we apply the same analysis
of the operator Θ− as was made in the case of a collision of the light
particle with ∂D, and then combine it with (B.10) and obtain the
bound
√
√
2ε̃0 kdv0 k/M +ε̃0 (1+C/ M ) kΘ− (dq1− )k/M ≤ 2ε̃0 (1+C/ M ) kdv1 k/M
Combining all these bounds gives
√
kdv1 −dv10 k ≤ (ε0 + 2ε̃0 /M ) (1+C/ M ) kdv1 k+C(δ0 +δ1 ) kdv1 k/ cos ϕ1
Lastly, we consider the vectors dV and dV 0 . These do not change
between collisions or due to collisions of the light particle with ∂D.
At an interparticle collision, we have, in a way similar to the previous
estimates
kdV1 − dV10 k ≤ kdV0 − dV00 k + 2kdv0 − dv00 k/M
+Cδ1 kdv0 k/M + Cδ1 kdq1− k/M
0 −
+kΘ− k kdq1− − (dq10 )− k/M + kΘ− k kdQ−
1 − (dQ1 ) k/M
(B.11)
+Cδ1 kΘ− k kdq1− k/(M cos ϕ1 )
Applying (B.5) and doing the same analysis of the operator Θ− as
before gives
√
kΘ− k kdq1− − (dq10 )− k/M ≤ ε0 (1 + C/ M ) kΘ− (dq1− )k/M
+C(δ0 + δ1 ) kdq1− k/(M cos ϕ1 )
BROWNIAN BROWNIAN MOTION
93
The first term on the right hand side can be combined with the term
2kdv0 − dv00 k/M in (B.11), and we get
√
A00 : = 2kdv0 − dv00 k/M + ε0 (1 + C/ M ) kΘ− (dq1− )k/M
√
≤ 2ε0 kdv0 k/M + 2ε0 (1 + C/ M ) kΘ− (dq1− )k/M
√
≤ 2ε0 (1 + C/ M ) kdv1 k/M
We collect the above estimates and obtain
√
kdV1 − dV10 k ≤ (2ε0 + ε̃0 ) (1 + C/ M ) kdv1 k/M
+C(δ0 + δ1 ) kdv1 k/(M cos ϕ1 )
Lemma B.1 is proved.
Remark. By fixing Q and taking the limit M → ∞ we obtain a version
of the above lemma for the billiard map FQ . It becomes much simpler,
of course, since dQi = dVi = 0 and ε̃i = 0, so we only get
δ0 + δ 1
ε1 = ε 0 + C
cos ϕ1
It is important to note that the constant C is uniform over all r >
0, as long as the points xi and x0i belong to the same u-curve or scurve (see Section 3.2), which can be verified directly. For example,
if q1 ∈ ∂P(Q), then |r1 − rr0 | ≤ cr|ϕ1 − ϕ01 | for some c > 0, hence
|n(q1 ) − n(q2 )| ≤ r−1 |r1 − rr0 | ≤ c|ϕ1 − ϕ01 | < cδ1 , which allows us to
suppress the large factor r−1 . The resulting distortion and curvature
bounds will be uniform over all r > 0.
Corollary B.2. Suppose that the point x1 in Lemma B.1 belongs to an
H-curve W1 . In that case the key estimates (B.3)–(B.4) of Lemma B.1
can be modified as follows:
2ε̃0
C
δ0 + δ 1
(B.12)
ε 1 = ε0 +
1+ √
+C
M
|W1 |2/3
M
C
δ0 + δ 1
(B.13)
ε̃1 = (2ε0 + ε̃0 ) 1 + √
+C
|W1 |2/3
M
with some constant C > 1 (possibly different from the one in Lemma B.1).
Proof. Indeed, if the points x1 and x01 lie in a homogeneity section Ik ,
then
|W1 | ≤ Const · [k −2 − (k + 1)−2 ] ≤ Const · (cos ϕ1 )3/2
This proves Corollary B.2.
94
N. CHERNOV AND D. DOLGOPYAT
We now extend the estimates made in Lemma B.1 and Corollary B.2
to an arbitrary iteration. We will show that the tangent vectors dxn
and dx0n are (εn , ε̃n )-close with some εn and ε̃n that we will estimate.
For brevity, put ei = (εi , ε̃i )T for i ≤ n. The bounds in Lemma B.1
and Corollary B.2 can be rewritten in matrix form
(B.14)
ei = Aei−1 + bi
where
C
1 2/M
A= 1+ √
2
1
M
is a fixed matrix and bi = (bi , bi )T , where bi = C (δi−1 + δi )/ cos ϕi
or bi = C (δi−1 + δi )/|Wi |2/3 depending on whether we want to apply
(B.3)–(B.4) or (B.12)–(B.13).
Iterating (B.14) gives
n
(B.15)
e n = A e0 +
n
X
An−i bi
i=1
√
1 2/M
The matrix
has eigenvalues λ1 = 1 + 2/ M and λ2 =
2
1
√
1 − 2/ M . By using its eigenvectors, we find
√ k
k
1
λk1 + λ√
(λk1 − λk2 )/ M
1 2/M
2
=
2
1
λk1 + λk2
(λk1 − λk2 ) M
2
We only need to apply our estimates to n ≤ M a with some a < 1/4,
according to Proposition 2.2 (b). Under this condition, for all k ≤ n
√ 1 0
k
A =
+ O k2 / M
2k 1
where the√
last term is a matrix whose entries are positive and less than
2
Const k / M . This bound and the expansion (B.15) imply
Lemma B.3 (n step distortion control). Assume that the standard
unstable vectors dx0 and dx00 are (ε0 , ε̃0 )-close for some small ε0 , ε̃0 > 0.
Then their images dxn and dx0n are (εn , ε̃n )-close with
εn = 2ε0 + ε̃0 + 2
n
X
bi
i=1
and
ε̃n = 3nε0 + 2ε̃0 + 3
n
X
i=1
(n − i)bi
BROWNIAN BROWNIAN MOTION
95
Now we are ready to prove Propositions 3.6 and 3.7. For brevity, we
will say ε-close instead of (ε, ε)-close.
Lemma B.4. Under the assumptions of Proposition 3.6, for any c > 0
there is a C > 0 such that whenever the tangent vectors dx0 and dx00
are (c|W0 (x0 , x00 )|/|W0 |2/3 )-close, then the tangent vectors dxi and dx0i
are (C|Wi (xi , x0i )|/|Wi |2/3 )-close for all i = 1, . . . , n.
Proof. Since the points xi and x0i belong to one H-curve, we can redefine δi to be |Wi (xi , x0i )|, and all our previous estimates will hold
(with maybe different values of the constants). Next, since δ0 ≤ |W0 |,
1/3
the initial tangent vectors dx0 and dx00 are (cδ0 )-close. Similarly, we
1/3
have δi−1 + δi ≤ Const δi , hence bi ≤ Const δi in the notation of
Lemma B.3. Recall that H-curves grow, locally, at an exponential rate
by (3.6), hence δi ≤ θ n−i δn for all i < n. We now employ Lemma B.3
1/3
and easily obtain that the tangent vectors dxn and dx0n are (Cδn )-close
with some C > 0. Therefore
J(x0 ) · · · J(xn−1 ) 1/3
(B.16)
ln J(x0 ) · · · J(x0 ) ≤ Cδn
0
n−1
for some C > 0. This estimate can be regarded as a weak distortion
bound, as compared to Proposition 3.6. But it provides us, at least,
with a uniform bound on distortions in the sense of (3.15) with some
β̃ > 0.
The exponential growth of H-curves (3.6) and the uniform bound
(3.15) imply that
n−i
δn
δi
3
≤
Cθ
|Wi |2/3
|Wn |2/3
for all i < n and some constant C > 0. Now we apply (B.14)–(B.15)
with bi = Const δi /|Wi |2/3 and easily obtain that the tangent vectors
dxn and dx0n are (Cδn /|Wn |2/3 )-close with some C > 0. Lemma B.4 is
proved.
Now Propositions 3.6 and 3.7 follow directly.
0
00
Finally, we prove Lemma 3.20. Let dy and dy be tangent vectors
to the curve γ at the points y 0 and y 00 , respectively. According to
the definition of standard pairs, we can assume that they are ε-close
with ε = Cdist(y 0 , y 00 )/|γ|2/3 . Then by Proposition 3.6, which we just
proved, the vectors dx0− = DF i (dy 0 ) and dx00− = DF i (dy 00 ) are ε− -close
with
ε− = Cdist(x0− , x00− )/|W |2/3 ≤ Const εγ /|W |2/3
We now compare the tangent vectors dx0 = (DFQ ◦ Dπ0 )(dx0− ) and
dx00 = (Dπ0 ◦ DF )(dx00− ).
96
N. CHERNOV AND D. DOLGOPYAT
Claim. dx0 and dx00 are ε0 -close with
Cεγ
Cεγ
(B.17)
ε0 =
+
|W |2/3 |W00 |2/3
where C > 0 is a large constant.
Proof follows the same lines as the proofs of Lemma B.1 and Corollary B.2, and we only focus on the novelty of the present situation.
First, since dist(x0− , x00− ) = O(εγ ) and dist(x0 , x00 ) = O(εγ ), then both
δ0 and δ1 in (B.12)–(B.13) will be O(εγ ). In addition, we apply Dπ0 to
both vectors. Recall that the projection π0 : Ω → Ω0 , fixes the position
of the heavy particle, sets its velocity to zero, and normalizes the vector w defined by (1.4). Accordingly, Dπ0 sets the components dQ and
dV of the tangent vector to zero and rescales the component dw in the
same way as w itself, i.e. it divides dw by kwk. In addition, we need
to project both components dq and dw onto the line perpendicular to
w, so that the basic equations (3.4)–(3.5) would hold. Therefore, the
map DπQ : (dQ, dV, dq, dw) 7→ (dQ1 , dV1 , dq1 , dw1 ) acts according to
the following rules: dQ1 = dV1 = 0, and
dw1 = dw/kwk − hdw, wi/kwk3,
dq1 = dq − hdq, wi/kwk2,
As we noted in Section 3.1, the estimates (a)-(g) of Proposition 3.1
apply to the vectors w and dw, instead of v and dv. Hence
kdq1 − dqk ≤ CkV k kdqk ≤ Cεγ kdqk
and
kwk dw1 − dw ≤ CkV k kdwk ≤ Cεγ kdwk
Such a difference can be incorporated into the right hand side of (B.17).
The division of dw by kwk results in a change of order one, in general,
but this will be matched by the corresponding division by kwk when
Dπ0 is applied to the other vector, as one can easily verify. This completes the proof of the claim.
We now finish the proof of Lemma 3.20. For each r ≥ 1 we need to
compare the tangent vectors dx0r = DFQr (dx0 ) and dx00r = DFQr (dx00 ).
The map FQ on ΩQ corresponds to the motion of the light particle when
the heavy one is fixed at Q, which is the limit case of our two-particle
system when M → ∞. Thus, our analysis in Appendix B, in particular
Lemma B.1, Corollary B.2, and Lemma B.3 apply to the map FQ as well
(without any restrictions on n, because M = ∞). In order to use them,
though, we need to verify the conditions they assume. First, since for
each r ≥ 1 the points FQr (x0 ) and FQr (x00 ) belong to one homogeneous
BROWNIAN BROWNIAN MOTION
97
stable manifold, they lie in one homogeneity section. Second, (B.1)–
(B.2) hold due to (3.13). Now Corollary B.2 and Lemma B.3 can be
used, indeed, and they directly imply Lemma 3.20.
References
[1] Bakhtin V. I. A direct method for constructing an invariant measure on a
hyperbolic attractor, Russian Acad. Sci. Izv. Math. 41 (1993) 207–227.
[2] Bakhtin, V. I. On the averaging method in a system with fast hyperbolic
motions, Proc. Belorussian Math. Inst. 6 (2000) 23–26.
[3] Bernstein S. N. Sur l’extension du theoreme limite de calcul des probabilites
aux sommes de quantites dependentes, Math. Ann. 97 (1926) 1–59.
[4] Bonetto F., Daems D. & Lebowitz J. Properties of stationary nonequilibrium
states in the thermostated periodic Lorenz gas I: the one particle system, J.
Stat. Phys. 101 (2000) 35–60.
[5] Bonetto F., Lebowitz J. & Rey-Bellet L. Fourier Law: a challenge to theorists,
in Mathematical Physics-2000, Imperial College Press, London, 2000.
[6] Bunimovich, L. A. On the ergodic properties of nowhere dispersing billiards,
Comm. Math. Phys. 65 (1979), 295–312.
[7] Bunimovich, L. A. & Sinai Ya. G. Statistical properties of Lorentz gas with
periodic configuration of scatterers, Comm. Math. Phys. 78 (1980/81) 479–
497.
[8] Bunimovich, L. A.; Sinai, Ya. G. & Chernov, N. I. Markov partitions for twodimensional hyperbolic billiards, Russian Math. Surveys bf 45 (1990) 105–152.
[9] Bunimovich L. A.; Sinai, Ya. G. & Chernov, N. I. Statistical properties of twodimensional hyperbolic billiards, Russian Math. Surveys 46 (1991) 47–106.
[10] Bunimovich L., Liverani C., Pellegrinotti A., Suhov Y. Ergodic systems of n
balls in a billiard table, Comm. Math. Phys. 146 (1992), 357–396.
[11] Bunimovich, L. A.; Spohn, H. Viscosity for a periodic two disk fluid: an existence proof, Comm. Math. Phys. 176 (1996), 661–680.
[12] Calderoni P., Durr D. & Kusuoka S. A mechanical model of Brownian motion
in half-space, J. Statist. Phys. 55 (1989), 649–693.
[13] Chernov N., Statistical properties of the periodic Lorentz gas. Multidimensional
case, J. Statist. Phys. 74 (1994), 11–53.
[14] Chernov N., Entropy, Lyapunov exponents and mean-free path for billiards, J.
Statist. Phys. 88 (1997), 1–29.
[15] Chernov, N. I. Markov approximations and decay of correlations for Anosov
flows, Ann. of Math. 147 (1998), 269–324.
[16] Chernov N., Decay of correlations in dispersing billiards, J. Statist. Phys. 94
(1999), 513–556.
[17] Chernov N., Sinai billiards under small external forces, Annales H. Poincare 2
(2001), 197–236.
[18] Chernov N. & Dettmann C. P. The existence of Burnett coefficients in the
periodic Lorenz gas, Phys. A 279 (2000) 37–44.
[19] Chernov N., Lebowitz J., and Sinai Ya. Scaling dynamic of a massive piston in
a cube filled with ideal gas: Exact results, J. Stat. Phys. 109 (2002), 529–548.
[20] Chernov N., Lebowitz J., and Sinai Ya. Dynamic of a massive piston in an
ideal gas, Russ. Math. Surveys 57 (2002), 1–84.
98
N. CHERNOV AND D. DOLGOPYAT
[21] Chernov N., and Lebowitz J. Dynamics of a massive piston in an ideal gas:
Oscillatory motion and approach to equilibrium, J. Statist. Phys. 109 (2002),
507–527.
[22] Cogburn R. & Ellison J. A. A four-thirds law for phase randomization of
stochastically perturbed oscillators and related phenomena, Comm. Math. Phys.
166 (1994), 317–336.
[23] Dolgopyat D. On differentiability of SRB states for partially hyperbolic systems,
preprint.
[24] Dolgopyat D. Limit Theorems for partially hyperbolic systems, to appear in
Trans. AMS.
[25] Durr D., Goldstein S. & Lebowitz J. L. A mechanical model of Brownian
motion, Comm. Math. Phys. 78 (1980/81), 507–530.
[26] Durr D., Goldstein S. & Lebowitz J. L. A mechanical model for the Brownian
motion of a convex body, Z. Wahrsch. Verw. Gebiete 62 (1983) 427–448.
[27] Field M., Melbourne I. & Torok A. Decay of correlations, central limit theorems
and approximation by Brownian motion for compact Lie group extensions, Erg.
Th. & Dyn. Syst. 23 (2003) 87-110.
[28] Freidlin M. I. & Wentzell A. D. Random perturbations of dynamical systems,
Grundlehren der Mathematischen Wissenschaften 260 (1984) Springer-Verlag,
New York.
[29] Hard Ball Systems and the Lorentz Gas, Edited by D. Szász. Encyclopaedia of
Mathematical Sciences, 101 Springer-Verlag, Berlin, 2000.
[30] Hartman P. Ordinary differential equations, 2d ed., 1982 Birkhauser, BostonBasel-Stuttgart.
[31] Ibragimov I. A. & Linnik Yu. V. Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971.
[32] Katok A., Knieper G., Pollicott M. & Weiss H. Differentiability and analyticity
of topological entropy for Anosov and geodesic flows, Inv. Math. 98 (1989) 581–
597.
[33] Khasminskii R. Z. Diffusion processes with a small parameter, Izv. Akad. Nauk
SSSR Ser. Mat. 27 (1963) 1281–1300.
[34] Kifer Yu. The exit problem for small random perturbations of dynamical systems with a hyperbolic fixed point, Israel J. Math. 40 (1981), 74–96.
[35] Kifer Yu. Limit theorems in averaging for dynamical systems, Erg. Th. & Dyn.
Sys. 15 (1995) 1143–1172.
[36] Kifer Yu. Averaging principle for fully coupled dynamical systems and large
deviations, to appear in Erg. Th. & Dyn. Sys.
[37] Krámli A., Simányi N., Szász D. A “transversal” fundamental theorem for
semi-dispersing billiards, Comm. Math. Phys. 129 (1990), 535–560.
[38] Liverani C. On contact Anosov flows, preprint.
[39] Liverani C. & Wojtkowski M., Ergodicity in Hamiltonian systems, in: Dynamics reported, 130–202, Dynam. Report. Expositions Dynam. Systems (N.S.),
4, Springer, Berlin, 1995.
[40] Machta J., Power Law Decay of Correlations in a Billiard Problem, J. Statist.
Phys. 32 (1983), 555–564.
[41] Melbourne I. & Török A., Statistical limit theorems for suspension flows, manuscript.
BROWNIAN BROWNIAN MOTION
99
[42] Pardoux E.& Veretennikov A. Yu. On the Poisson equation and diffusion
approximation-I, Ann. Probab. 29 (2001), 1061–1085.
[43] Parry W. & Pollicott M. Zeta functions and the periodic orbit structure of
hyperbolic dynamics, Asterisque 187-188 (1990).
[44] Pene F. Averaging method for differential equations perturbed by dynamical
systems, ESAIM Probab. Stat. 6 (2002), 33–88 (electronic).
[45] Pesin Ya. B. & Sinai Ya. G. Gibbs measures for partially hyperbolic attractors,
Erg. Th. & Dyn. Sys. 2 (1982) 417–438.
[46] Revuz D. & Yor M. Continuous Martingales and Brownian Motion, 3d ed.,
1998 Sringer, Berlin-Heildelberg-New York.
[47] Ruelle D. Invariant measures for a diffeomorphism which expands the leaves
of a foliation, Publ. IHES. 48 (1978) 133–135.
[48] Ruelle D. Differentiation of SRB states, Comm. Math. Phys. 187 (1997) 227–
241.
[49] Shub M. Global stability of dynamical systems, With the collaboration of Albert
Fathi and Remi Langevin, Springer-Verlag, New York, 1987.
[50] Simányi N. Ergodicity of hard spheres in a box, Ergod. Th. Dynam. Syst. 19
(1999), 741–766.
[51] Simányi, N. & Szász, D. Hard ball systems are completely hyperbolic, Annals
of Math. 149 (1999), 35–96.
[52] Sinai, Ya. G. Classical dynamic systems with countably-multiple Lebesgue spectrum: I,II, Izv. Akad. Nauk SSSR Ser. Mat. 25 (1961) 899–924, 30 (1966)
15–68.
[53] Sinai, Ya. G. Dynamical systems with elastic reflections: Ergodic properties of
dispersing billiards, Russ. Math. Surv. 25 (1970), 137–189.
[54] Soloveitchik M. Mechanical background of Brownian motion, in Sinai’s Moscow
Seminar on Dynamical Systems, AMS Transl. Ser. 2, 171 AMS , Providence,
RI, 1996. 233–247.
[55] Spohn H. Large scale dynamics of interacting particles, Springer-Verlag, Berlin,
New York, 1991.
[56] Stroock D. W. & Varadhan S. R. S. Multidimensional diffusion processes,
Grundlehren der Mathematischen Wissenschaften 233, (1979) SpringerVerlag, Berlin-New York.
[57] Young L.–S. Statistical properties of dynamical systems with some hyperbolicity, Ann. Math. 147 (1998) 585–650.