DISSIPATION TIME AND DECAY OF CORRELATIONS
ALBERT FANNJIANG†, STÉPHANE NONNENMACHER‡ AND LECH WOLOWSKI†
Abstract. We consider the effect of noise on the dynamics generated by volume-preserving maps
on a d-dimensional torus. The quantity we use to measure the irreversibility of the dynamics is the
dissipation time. We focus on the asymptotic behaviour of this time in the limit of small noise. We
derive universal lower and upper bounds for the dissipation time in terms of various properties of
the map and its associated propagators: spectral properties, local expansivity, and global mixing
properties. We show that the dissipation is slow for a general class of non-weakly-mixing maps; on
the opposite, it is fast for a large class of exponentially mixing systems which include uniformly
expanding maps and Anosov diffeomorphisms.
1. Introduction
The origin of irreversibility in dynamical systems is often modeled by small stochastic perturbations of the otherwise reversible dynamics. These perturbations may be attributed to uncontrolled
interactions with the ‘environment’, or with internal variables neglected in the equations. In experimental or numerical investigations, stochasticity or ‘noise’ is introduced respectively by finite
precision of the preparation and measurement apparatus, and by rounding-off errors due to finite
computer precision.
One may take these stochastic perturbations explicitly into account by adding a term of Langevin
type in the evolution equations, or equivalently introducing some diffusion term in the FokkerPlanck equation. In the present article we choose to deal with discrete-time dynamics on some
compact phase space, namely the d-dimensional torus. The maps we study conserve the volume
element (or Lebesgue measure). The noise is implemented through a convolution operator, the
kernel of which has a (small) width > 0. For a given map F , two types of stochastic perturbations
will be considered: the noise operator may be applied at each step of evolution, resulting in a
“noisy evolution”; on the opposite, we may choose to introduce some stochasticity only at the
initial (preparation) and final (measurement) steps, resulting in a “coarse-grained evolution”.
In general, the influence of the noise on the long-time evolution of the system depends on the
dynamical properties of the map. Typically, the effect is stronger if the map is ‘chaotic’ than
if it is ‘regular’ [?, ?, ?]. We will put this statement into quantitative estimates. In this aim,
we will characterize the effect of noise on a map F through a single observable, the dissipation
time. By dissipation, we mean the damping (through the noisy or coarse-grained evolution) of the
density fluctuations, while the total probability remains constant. Mathematically, this damping is
expressed by the decay of the L2 -norm of the density (or equivalently, the L2 -norm of the density
fluctuations). The decrease of this norm due to dissipation may be interpreted as an “entropy
increase” of the system, up to the state of maximal entropy (i.e. the uniform density). We define
the dissipation time as the time needed for the evolution to bring fluctuations under a fixed threshold
(that is, reduce the L2 norm by a fixed factor) [?]. We are especially interested in the behaviour of
this time in the limit of small noise. We will show that this behaviour drastically depends on the
† Department of Mathematics, University of California at Davis, Davis, CA 95616, USA (email:
[email protected], [email protected]). The research of AF is supported in part by the grant
from U.S. National Science Foundation, DMS-9971322 and UC Davis Chancellor’s Fellowship.
‡ Service de Physique Théorique, CEA/DSM/PhT (Unité de recherche associée au CNRS) CEA/Saclay 91191
Gif-sur-Yvette cédex, France ([email protected]).
1
dynamics generated by the noiseless map F : in short, the dissipation will be “fast” for a chaotic
dynamics, as opposed to “slow” for a regular dynamics, the difference being ever more striking as
the level of noise decreases. Our main results will be quantitative estimates of the dissipation time,
which emphasize this opposition.
Since the dissipation time is defined in terms of an L2 norm, it is naturally related to the
spectral properties of the propagator associated with the map, acting on the space of squareintegrable functions. In Section ??, we analyze the links between, on one side, the spectrum and
pseudospectrum of the noisy or noiseless propagators, on the other side, their dissipation time.
The relevance of the pseudospectrum for time evolution problems has been recently put in evidence
in the context of non-unitary continuous-time dynamical systems [?]. These spectral relationships
will be mainly used to analyze the case of “regular”, precisely non-weakly-mixing maps. On the
opposite, they are of little help for more chaotic ones.
In the following Sections, we connect the dissipation time to more ‘dynamical’ properties of the
map F (under some smoothness assumptions on F ). We first obtain lower bounds for the dissipation
time from the local expansion properties of the map: a weak, or absence of local expansion will
imply a slow dissipation (Section ??). Then, we obtain upper bounds for the dissipation time using
informations on the mixing properties of the map (that is, the time decay of correlations between
observables). This mixing is a typical characteristics of chaotic behaviour, it can be easily measured
in numerical simulations [?]. The main conclusion is that exponential or stronger mixing implies
fast dissipation, for both the noisy and the coarse-grained evolution.
In Section ??, we describe several families of volume-preserving maps on the torus, for which the
results obtained above yield useful information concerning the dissipation. The main two families
are, on the one hand, expanding (noninvertible) maps, on the other hand, Anosov (or at least
partially hyperbolic) diffeomorphisms of the torus. All these maps are mixing, so the dissipation
is “fast”. In the case of Anosov maps, we collect our results in Theorem ??. As a matter of
illustration, we also analyze in detail examples of linear mixing maps, for which exact asymptotics
of the dissipation times can be obtained, and therefore give an idea of the sharpness of the bounds
obtained before (the results concerning linear automorphisms had been obtained in [?]).
From these developments one notices that dissipation time provides a more robust characteristic of the system than the decay rate of dynamical correlation. E.g. the former has the same
asymptotic, up to a constant factor, among all the Anosov diffeomorphisms while the latter has the
exponential asymptotics for generic Anosov diffeomorphisms but a super-exponential asymptotic
for the hyperbolic toral automorphisms (cf. Corollary ??).
2. Setup and notation
2.1. Evolution operators. Let (Td , B(Td ), m) denote the d-dimensional torus, equipped with its
σ−field of Borel sets and the Lebesgue measure m. Let F : Td → Td be a map on the torus
preserving the Lebesgue measure: for any set B ∈ B(Td ) we have m(F −1 (B)) = m(B). In general,
F is not supposed to be invertible. In the following we call such a map ‘volume preserving’ with
implicit reference to the Lebesgue measure.
The map F generates a discrete time dynamics on Td , which in terms of pathwise description can
be represented by the forward trajectory {F n (x0 ), n ∈ N} of any initial point (particle) x0 ∈ Td .
However, instead of looking at the evolution of a single particle, one can consider the statistical
description of the dynamics, that is the evolution of a density (more generally a measure) describing
the initial statistical configuration of the system.
Let M(Td ) denote the set of all Borel measures on Td . For any µ ∈ M(Td ) and f ∈ C 0 (Td ) we
write
Z
µ(f ) =
f (x)dµ(x).
Td
2
The map F induces a map F ∗ on M(Td ) given by
(F ∗ µ)(f ) = µ(f ◦ F ),
for all f ∈ C 0 (Td ).
This map can also be defined as follows:
(F ∗ µ)(B) = µ(F −1 (B)),
for all B ∈ B(Td ).
In particular if µ = δx0 then F ∗ (µ) = δF (x0 ) and one recovers the pathwise description.
If µ is absolutely continuous w.r.t. m, then F ∗ (µ) preserves this property (since the measuredµ
preserving map F is nonsingular w.r.t. m, see [?, p.42]). The corresponding densities g = dm
∈
L1 (Td ) are transformed by the Frobenius-Perron or transfer operator PF [?]:
dµ
d(F ∗ µ)
=
.
PF
dm
dm
If the map F is invertible, PF is given explicitly by:
dF ∗ m
(x) = g ◦ F −1 (x).
dm
If the map F is differentiable, and the preimage set of x is a finite set for all x, the Perron-Frobenius
operator is given by
X
g(y)
(PF g)(x) =
,
|JF (y)|
(PF g)(x) = (g ◦ F −1 )(x)
y|F (y)=x
where JF (y) is the Jacobian of F at y.
On the other hand one can consider the dual of the Frobenius-Perron operator, called the Koopman operator, which governs the evolution of observables f ∈ L∞ (Td ) instead of that of densities
g ∈ L1 . The Koopman operator UF is defined as
(1)
UF f = f ◦ F.
Due to the nonseparability of the Banach space L∞ (Td ), it is often more convenient to consider its
closure in some weaker Lp norm, which yields larger (but separable) spaces of observables Lp (Td ).
In this paper we will be mainly concerned with the space L2 (Td ) and its codimension-1 subspace
of zero-mean functions L20 (Td ) = {f ∈ L2 (T2 ) : m(f ) = 0}. This subspace is obviously invariant
through UF or PF , due to the assumption F ∗ m = m. Throughout the paper, k · k will always refer
to the L2 -norm (and corresponding operator norm) on L20 (Td ), deriving from the Hermitian scalar
product h·, ·i (any other norm will carry an explicit subscript).
For any measure-preserving map F , the operator UF is isometric on L2 (T2 ) and L20 (Td ). When
F is invertible, UF is unitary on these spaces, and satisfy UF = PF−1 = PF −1 .
2.1.1. Additional notation. Although the operators introduced in previous sections were mostly
defined on the space L20 (Td ), it will be useful to consider other function spaces, which we define
now in some detail. For any m ∈ N, we denote by C m (Td ) the space of m-times continuously
differentiable functions, with the norm
X
kf kC m =
kDα f k∞
|α|1 ≤m
(we use the norm |α|1 = α1 + . . . + αd for the multiindex α ∈ Nd ). For any s = m + η with
m = [s] ∈ N, η ∈ (0, 1), let C s (Td ) denote the space of C m functions for which the m-derivatives
are η-Hölder continuous; this space is equipped with the norm
X
|Dα f (x) − Dα f (y)|
kf kC s = kf kC m +
sup
|x − y|η
x6=y
|α|1 =m
3
The Fourier transforms of functions g ∈ L1 (Rd ) and f ∈ L1 (Td ) are defined as follows:
Z
d
(2)
∀ξ ∈ R , ĝ(ξ) =
g(x)e−2πix·ξ dx,
Rd
Z
d
ˆ
(3)
f (x)e−2πix·k dx = hek , f i.
∀k ∈ Z , f (k) =
Td
Above we used the Fourier modes on the torus ek (x) := e2πix·k . For any s ≥ 0, we denote by
H s (Td ) and H s (Rd ) the Sobolev spaces of s-times weakly differentiable L2 -functions equipped with
the norms k · kH s defined respectively by
Z
2
kgkH s (Rd ) =
(1 + |ξ|2 )s |ĝ(ξ)|2 dξ,
ξ∈Rd
kf k2H s (Td )
=
X
(1 + |k|2 )s |fˆ(k)|2 .
k∈Zd
Finally, for any of these spaces, adding the subscript 0 will mean that we consider the (UF -invariant)
subspace of functions with zero average, e.g. C0j (Td ) = {f ∈ C j (Td ), m(f ) = 0}.
2.2. Noise operator. To construct the noise operator we consider an arbitrary symmetric probability density function g ∈ L1 (Rd ), called the noise generating density. The noise width (or noise
level ) will be given by a single nonnegative parameter, which we call . To each > 0 corresponds
the noise kernel on Rd :
1 x
,
g (x) = d g
with the convention that g0 = δ0 . The noise kernel on the torus is obtained by periodizing g ,
yielding the periodic kernel
X
(4)
g̃ (x) =
g (x + n).
n∈Zd
We remark that the Fourier transform of g̃ is related to that of g by the identities g̃ˆ (k) = ĝ (k) =
ĝ(k).
The noise operator G is defined on any function f ∈ L20 (Td ) as the convolution:
G f
= g̃ ∗ f.
L1
As a convolution operator defined by an
density, G is compact on L20 (Td ) (if g is squareintegrable, G is Hilbert-Schmidt). The Fourier modes {ek , k ∈ Zd \ {0}} form an orthonormal
basis of eigenvectors of G , yielding the following spectral decomposition:
X
(5)
∀f ∈ L20 (Td ), G f =
ĝ(k)hek , f i ek .
06=k∈Zd
This formula shows that the eigenvalue associated with ek is ĝ(k). Since g is a symmetric function,
this eigenvalue is real, so that G is a self-adjoint operator. Its spectral radius rsp (G ) is therefore
given by
(6)
rsp (G ) = kG k = sup |ĝ(k)|.
06=k∈Zd
Since the density g is positive, ĝ attains its maximum ĝ(0) = 1 at the origin and nowhere else.
Besides, because g ∈ L1 (Rd ), ĝ is a continuous function vanishing at infinity. As a result, for small
enough > 0, the supremum on the RHS of (??) is reached at some point k close to the origin,
and this maximum is strictly smaller than 1. This shows that the operator G is strictly contracting
on L20 (Td ):
(7)
∀ > 0,
kG k = rsp (G ) < 1.
4
In the next section we study this noise operator more precisely, starting from appropriate assumptions on the noise generating density.
2.3. Noise kernel estimates. In this subsection we present some estimates regarding the noise
operator, which will be used throughout the paper to estimate the dissipation time.
We will be interested in the behaviour of the system in the limit of small noise level, that is the
limit → 0. It will hence be useful to introduce the following asymptotic notation. Given two
variables a , b depending on > 0, we write
a
(8)
a . b if lim sup
< ∞,
b
→0
a
(9)
a ≈ b if lim
= 1,
→0 b
(10)
a ∼ b if a . b and b . a .
In order to obtain interesting estimates on the noise operator G , it will be necessary to impose
some additional conditions on its generating density g, regarding e.g. its rate of decay at infinity,
or the behaviour of its Fourier transform near the origin.
The weakest condition considered in this paper is the existence of some positive moment of g,
by which we mean that for some α ∈ (0, 2],
Z
(11)
Mα =
|x|α g(x)dx < ∞
Rd
(we take the length |x| = (x21 + . . . + x2d )1/2 on Rd ). This condition implies the following properties
of the Fourier transform ĝ (proved in Appendix ??):
Lemma 1. For any α ∈ (0, 2] there exists a universal constant Cα such that, if a normalized
density g satisfies (??), then the following inequalities hold:
(12)
∀ξ ∈ Rd ,
0 ≤ 1 − ĝ(ξ) ≤ Cα Mα |ξ|α .
If (??) holds with α = 2, we have the more precise behaviour:
1 − ĝ(ξ) ∼ |ξ|2
in the limit ξ → 0.
In the case α < 2, we will sometimes assume a stronger property than (??), namely that
(13)
1 − ĝ(ξ) ∼ |ξ|α
in the limit ξ → 0.
Notice that this behaviour implies a uniform bound 1 − ĝ(ξ) ≤ C|ξ|γ for any γ ≤ α and C
independent of γ.
Typical examples of noise kernels satisfying (??) include the Gaussian kernel and more general
symmetric α−stable kernels [?, p.152] defined for α ∈ (0, 2]:
X
α/2
(14)
g,α (x) :=
e−(Q(k)) ek (x),
k∈Zd
where Q denotes an arbitrary positive definite quadratic form. For the values of α indicated, the
function g,α (x) is positive on Rd .
In view of Eq.(??), the properties (??) or (??) directly constraint the rate at which G contracts
on L20 (Td ). For instance, (??) implies that in the limit → 0,
(15)
1 − kG k ∼ α .
The following proposition describes the effect of the noise on various types of observables, in the
limit of small noise level. The proofs are given in Appendix ??.
5
Proposition 1. i) For any noise generating density g ∈ L1 (Rd ) and any observable f ∈ L20 (Td ),
one has
→0
kG f − f k−−→0.
(16)
To obtain information on the speed of convergence, we need to impose constraints on both the noise
kernel and the observable.
ii) If for some α ∈ (0, 2] the kernel g satisfies (??) or (??), then for any γ > 0 there exists a
constant C > 0 such that for any observable f ∈ H γ (Td ),
(17)
kG f − f k ≤ Cγ∧α kf kH γ∧α ,
where γ ∧ α := min{γ, α}. If f ∈ C 1 (Td ), the above upper bound can be replaced by
(18)
kG f − f k ≤ C1∧α k∇f k ≤ C1∧α k∇f k∞ .
Using the noise operator, we are now in position to define the noisy (resp. the coarse-grained)
dynamics generated by a measure-preserving map F .
2.4. Noisy evolution operator and dissipation time. The noisy evolution through the map
F is constructed by successively applying the Koopman operator UF and the noise operator G ,
therefore by taking powers of the noisy propagator
T = G UF .
In general, the operator T is not normal, but satisfies rsp (T ) ≤ kT k = kG k.
We will also consider a coarse-grained dynamics defined by the application of the noise kernel only
at the beginning and the end of the evolution. Hence we define the following family of operators:
T̃(n) = G UFn G ,
n ∈ N.
(n)
In view of the contracting properties of G , the inequalities kTn k ≤ kG kn , kT̃ k ≤ kG k2
imply that both noisy and coarse-grained operators are strictly contracting on L20 (Td ). Our aim
is to characterize the speed of contraction of these two evolutions, that is the behaviours of the
(n)
norms kTn k, kT̃ k in the joint limits → 0 and n → ∞. This characterization will be connected
with dynamical properties of the map F .
There are many ways to measure this speed of contraction. For instance, for fixed > 0, the longtime decay of kTn k may be super-exponential (in which case T is quasinilpotent) or exponential
governed e.g. by the largest eigenvalue of T . However, such exponential behaviour may appear
only after a transient time. In this paper we will characterize the noisy dynamics through a single,
robust characteristics, namely the dissipation time. In its general form the dissipation time τ∗ is
defined in terms of the norm k · kp,0 on the space Lp0 (Td ), and an arbitrary threshold η ∈ (0, 1):
(19)
τ∗p,η () := min{n ∈ N : kTn kp,0 < η},
1 ≤ p ≤ ∞.
We will be concerned with the behaviour of the dissipation time when the level of noise becomes
small (as we prove in Proposition ??, this time diverges in this limit). In [?] it was shown that this
asymptotic behaviour is independent of the choice of 0 < η < 1 and 1 < p < ∞. Therefore, in the
present paper we will consider the computationally convenient choice p = 2 and η = e−1 . We will
henceforth drop the superscripts, and the dependence of τ∗ on will always be implicit:
(20)
−1
τ∗ := τ∗2,e () = min{n ∈ N : kTn k < e−1 }.
A similar dissipation time will be defined for the coarse grained evolution:
(21)
τ̃∗ := min{n ∈ N : kT̃(n) k < e−1 }.
6
The dissipation time does not depend on whether the dynamics is applied to densities (i.e. by
the Frobenius-Perron operator) or to observables (by the Koopman operator). Indeed, the norm of
an operator equals the norm of its adjoint [?, p.195], so that
kT̃(n) k = kG UFn G k = k(G UFn G )∗ k = kG PFn G k,
and similarly for the noisy operator T . In particular, for invertible maps the dissipation time does
not depend on the direction of time.
We will distinguish two qualitatively different asymptotic behaviours of dissipation time in the
limit → 0. We say that the operator T (or the map F associated with it) respectively has
I) simple or power-law dissipation time if there exists β > 0 such that
τ∗ ∼ 1/β ,
II) fast or logarithmic dissipation time if
τ∗ ∼ ln(1/).
We will also talk about slow dissipation time whenever there exists some β > 0 s.t.
τ∗ & 1/β .
In case of logarithmic dissipation time, the dissipation rate constant R∗ , when it exists, is defined
as
τ∗
(22)
.
R∗ = lim
→0 ln(1/)
A similar terminology will be applied to the coarse-grained dissipation time τ̃∗ .
3. Spectral properties and dissipation time of non-weakly-mixing maps
We investigate the connection between the dissipation time of the noisy propagator T , its pseudospectrum and some spectral properties of UF and G . All the operators considered in this section
are defined on L20 (Td ). In the framework of continuous-time dynamics, connections have been obtained between, on one side, the pseudospectrum of the (non-selfadjoint) generator A, on the other
side, the norm of the evolution operator etA [?]. We will obtain results of the same flavor, yet the
proofs seem here easier than in the case of continuous time.
3.1. Definitions and general bounds. Let us start with the definition of the pseudospectrum
of a bounded operator [?].
Definition 1. Let T be a bounded linear operator on a Hilbert space H (we note T ∈ L(H)). For
any δ > 0, the δ-pseudospectrum of T (denoted by σδ (T )) can be defined in the following three
equivalent ways:
(I) σδ (T ) = {λ ∈ C : k(λ − T )−1 k ≥ δ −1 },
(II) σδ (T ) = {λ ∈ C : ∃v ∈ H, kvk = 1, k(T − λ)vk ≤ δ},
(III) σδ (T ) = {λ ∈ C : ∃B ∈ L(H), kBk ≤ δ, λ ∈ σ(T + B)}.
We will apply these definitions to the operator T . For brevity, the resolvent of this operator will
be denoted by R (λ) = (λ − T )−1 . We call S r the circle {λ ∈ C : |λ| = r} in the complex plane,
and define the following pseudospectrum distance function:
d (r) := inf{δ > 0 : σδ (T ) ∩ S r 6= ∅}.
From the definition (I) of the pseudospectrum, one easily shows that this distance is also given by
(23)
d−1
(r) = sup kR (λ)k.
|λ|=r
7
We first establish general (abstract) bounds for the dissipation time in terms of the spectral
properties of G and T . In a second step, we relate these properties to dynamical properties of the
underlying map F .
Theorem 1. For any isometric operator U on L20 (Td ) and noise operator G , the dissipation time
of the noisy evolution operator T = G U satisfies the following estimates:
(24)
(25)
1
1 − e−1
≤ τ∗ ≤
+ 1,
d (1)
|ln(kG k)|
1
e
τ∗ ≤
inf
ln
.
d (r)
rsp (T )<r<1 | ln(r)|
We notice that the first upper bound does not depend on U at all, but only on the noise. Using
the estimate (??), we obtain the following obvious corollary:
Corollary 1. If the noise generating density satisfies the estimate (??) for some α ∈ (0, 2], then
for any measure-preserving map F the noisy dissipation time is bounded from above by:
τ∗ . −α .
Proof of the Theorem.
1. Lower bound
We use the following series expansion of the resolvent [?, p.211] valid for any |λ| > rsp (T ):
(26)
R (λ) =
∞
X
λ−n−1 Tn .
n=0
Considering that rsp (T ) ≤ kG k < 1, we may take |λ| = 1, and cut this sum into two parts:
R (λ) =
τX
∗ −1
λ−n−1 Tn + λ−τ∗ Tτ∗ R (λ).
n=0
Taking norms and applying the triangle inequality, we get
kR (λ)k ≤ k
τX
∗ −1
λ−n−1 Tn k + |λ|−τ∗ kTτ∗ kkR (λ)k
n=0
≤ τ∗ + e−1 kR (λ)k
=⇒ kR (λ)k(1 − e−1 ) ≤ τ∗ .
Taking the supremum over λ ∈ S1 yields the lower bound.
2. Upper bounds
To get both upper bounds, we use the following trivial lemma.
Lemma 2. Assume that (for some value of ) the powers of T satisfy
∀n ∈ N,
kTn k ≤ Γ(n),
n→∞
where the function Γ(n) is strictly decreasing, and Γ(n)−−−→0. Then the dissipation time is
bounded from above by
τ∗ ≤ Γ(−1) (e−1 ) + 1,
where Γ(−1) is the inverse function of Γ. In particular, for the geometric decay Γ(n) = Crn with
r ∈ (0, 1), C ≥ 1, one obtains τ∗ ≤ ln(eC)
| ln r| + 1.
8
The upper bound in Eq. (??) comes from the obvious estimate
kTn k ≤ kG kn ,
on which we apply the lemma with C = 1, r = kG k.
To prove the second upper bound, we use the representation of Tn in terms of the resolvent:
Z
1
Tn =
λn R (λ)dλ
2πi S r
valid for any r > rsp (T ). Thus for all r ∈ (rsp (T ), 1), one has
Z
1
1
kTn k ≤
|λ|n kR (λ)k|dλ| ≤ sup kR (λ)krn+1 =
rn+1 .
2π S r
d
(r)
|λ|=r
We then apply Lemma ?? on the geometric decay for any radius rsp (T ) < r < 1, with C =
r
d (r)
≥ 1.
3.2. Consequences. We now use Theorem ?? in the case where U = UF is the Koopman operator
for some measure-preserving map F on Td with some specific dynamical properties. We recall [?]
that the map F is ergodic (resp. weakly-mixing) iff 1 is not an eigenvalue of UF (resp. iff UF has
no eigenvalue) on L20 (Td ).
Proposition 2. For any measure-preserving map F and any noise generating function g, the
dissipation time of T diverges in the small-noise limit → ∞.
Proof. We skip the subscript F to alleviate notation. We only use the fact that U = UF is an
isometry. We shall prove by induction the following strong convergence of operators
∀f ∈ L20 (Td ),
→0
kTn f − U n f k−−→0.
∀n ∈ N,
From Proposition ??i), this limit holds in the case n = 1. Let us assume it holds at the rank n − 1.
Then we write
Tn f = U Tn−1 f + (G − I)U Tn−1 f.
→0
From the inductive hypothesis, Tn−1 f −−→U n−1 f , so that the first term on the RHS converges to
U n f . Applying Proposition ??i) to the function U n f , we see that the second term vanishes in the
→0
→0
limit → 0. From the isometry of U , we obtain that for any n > 0, kTn k−−→1, so that τ∗ −−→∞.
This “non-finiteness” of the noisy dissipation time allows us to prove another general result
concerning the pseudospectrum of T (this corollary is proved in the Appendix ??):
Corollary 2. For any isometry U and noise generating function g, one has
(27)
→0
d (1)−−→0.
This means that for any fixed δ > 0, the pseudospectrum σδ (T ) will intersect the unit circle for
small enough .
In order to better control the growth of τ∗ , we need more precise information on the noise and the
dynamics. In the present section, we restrict ourselves to the dynamical property of weak-mixing.
Corollary 3. Assume that the noise generating density g satisfies the estimates (??) or (??) with
exponent α ∈ (0, 2]. If F is not weakly-mixing and at least one eigenfunction of UF belongs to
H γ (Td ) for some γ > 0, then T has slow dissipation time:
−(α∧γ) . τ∗ .
9
Proof. Let h ∈ H γ (Td ) be a normalized eigenfunction of UF with eigenvalue λ. Applying
Proposition ??ii), we get
k(λ − T )hk = k(I − G )hk ≤ Kγ∧α
for some constant K > 0 depending on g and h. This implies that kR (λ)k ≥ K1γ∧α , therefore
taking the supremum over |λ| = 1 yields d (1)−1 ≥ K1γ∧α . The lower bound in Theorem ?? then
implies
(28)
1 − e−1
≤ τ∗ .
Kγ∧α
Remark 1. Recall that if g satisfies (??) with exponent α, then the dissipation time is also bounded
from above, as shown in Corollary ??. If one eigenfunction h has regularity H γ with γ ≥ α, then
both Corollaries imply that the dissipation is simple, of exponent α.
Remark 2. The above results can be stated in more general form: UF does not need to be a
Koopman operator associated with a map F . The result holds true for any isometric operator U on
L20 with an eigenfunction of Sobolev regularity.
The dependence of the lower bound in (??) on γ can be intuitively explained as follows. In case of
non-weakly-mixing maps the eigenfunctions of UF are, in general, responsible for slowing down the
dissipation. The rate of the dissipation is affected by the regularity of the smoothest eigenfunction.
In principle, irregular functions undergo faster dissipation giving rise to slower asymptotics of τ∗ .
It is not clear, however, whether the actual asymptotics of the dissipation time will be slower then
power law in case when all eigenfunctions of UF on L20 (Td ) are outside any space H γ (Td ) with
γ > 0.
In Corollary ?? we have shown that for any map F and arbitrary small δ > 0, the pseudospectrum
σδ (T ) intersects the unit circle for sufficiently small > 0. If F is not weakly-mixing, the spectral
radius of T (that is, the modulus of its largest eigenvalue) is believed to converge to 1 when → 0,
and the associated eigenstate h should converge to a “noiseless eigenstate” h. This “spectral
stability” has been discussed for several cases in the continuous-time as well as for discrete-time
maps on T2 [?, ?].
On the opposite, if F is an Anosov map on T2 (see Section ??), the spectrum of T does not
approach the unit circle, but stays away from it uniformly: rsp (T ) is smaller than some r0 < 1 for
any > 0 [?]. Simultaneously, kT k → 1, so we have here a clear manifestation of the nonnormality
of T for such a map. In some cases (see [?] and the linear examples of Section ??), the operator
T is even quasinilpotent, meaning that rsp (T ) = 0 for all > 0. For such an Anosov map, the
spectral radius of T is therefore “unstable” or “discontinuous” in the limit → 0, while in the
same limit the (radius of its) pseudospectrum σδ (T ) (for δ > 0 fixed) is “stable”.
4. Universal lower bounds for the dissipation time of C 1 maps
In this section we consider both noisy and coarse-grained evolutions. We start our discussion
with general properties of the coarse-grained dissipation time.
Proposition 3. Let F be a measure-preserving map.
i) For arbitrary noise kernel, the coarse-grained dissipation time diverges as → 0.
ii) If F is not weakly-mixing then τ̃∗ = ∞ for small enough > 0.
Proof.
i) Similarly as in Proposition ?? we have
kT̃n f − U n f k = kG UFn (G − I)f + (G − I)U n f k ≤ k(G − I)f k + k(G − I)U n f k → 0.
10
ii) Let h ∈ L20 (Td ) be a normalized eigenfunction of UF , then
kT̃(n) hk = kG UFn (h + (G − I)h)k ≥ kG hk − kG UFn (G − I)hk
≥ 1 − 2k(G − I)hk.
(n)
Since the RHS above is independent of n, we see that kT̃ k is close to 1 for all times and sufficiently
small > 0.
As opposed to the noisy case (see Prop. ??), the coarse-grained evolution through a non-weaklymixing map does not dissipate. Therefore there exists no general upper bound for τ̃∗ .
On the opposite, we will prove below a general lower bound for both coarse-grained and noisy
evolutions, valid for any measure-preserving map F of regularity C 1 . We note that Propositions ??
and ??i) (which are valid independently of any regularity assumption) do not provide an explicit
lower bound.
First we introduce some notation. For any map F ∈ C 1 , DF (x) is the tangent map of F at the
point x ∈ Td , mapping a tangent vector at x to a tangent vector at F (x). Selecting the canonical
(i.e. Cartesian) basis and metrics on T (Td ), this map can be represented as a d × d matrix. The
metrics naturally yields a norm v ∈ Tx (Td ) 7→ |v| on the tangent space, and therefore a norm
on this matrix: |DF (x)| = max|v|=1 |DF (x) · v|. We are now in position to define the maximal
expansion rate of F :
µF = lim sup kDF n k1/n
∞ ,
kDF n k∞ = sup |(DF n )(x)|.
where
n→∞
x∈Td
Since F preserves the Lebesgue measure, the Jacobian JF (x) satisfies |JF (x)| ≥ 1 at all points. In
the Cartesian basis, JF (x) = det(DF (x)), so the we have kDF n (x)k ≥ 1 for all x ∈ Td , n ≥ 0.
One can actually prove the following:
Remark 3. Although |(DF n )(x)| and kDF k∞ may depend on the choice of the metrics, µF does
not, and satisfies 1 ≤ µF ≤ kDF k∞ .
From the definition of µF , for any µ > µF there exists a constant A ≥ 1 such that
(29)
∀n ∈ N,
kDF n k∞ ≤ Aµn .
In some cases one may take µ = µF in the RHS. In case µF = 1, kDF n k∞ can sometimes grow as
a power-law:
(30)
kDF n k∞ ≤ Anβ ,
n∈N
for some β > 0, or even be uniformly bounded by a constant (β = 0).
The relationship between, on one side, the local expansion of the map F and on the other side,
the dissipation time, can be intuitively understood as follows. A lack of expansion (kDF k∞ = 1)
results in the transformation of “soft” or “long-wavelength” oscillations into “soft oscillations”,
both being little affected by the noise operator G . On the opposite, a locally strictly expansive
map (kDF k∞ > 1) will quickly transform soft oscillations into “hard” or “short-wavelength”, the
latter being much more damped by the noise.
The following theorem precisely measures this relationship, in terms of lower bounds for the
dissipation times.
Theorem 2. Let F be a measure-preserving C 1 map on Td , and assume that the noise generating
density g satisfies (??) or (??) for some α ∈ (0, 2].
i) If kDF k∞ > 1, resp. µF > 1, then there exist a constant c, resp. constants µ ≥ µF and c̃,
such that for small enough ,
α∧1
α∧1
(31)
τ∗ ≥
ln(−1 ) + c, resp. τ̃∗ ≥
ln(−1 ) + c̃.
ln(kDF k∞ )
ln µ
11
If F is a C 1 diffeomorphism, then (??) holds with kDF k∞ replaced by kDF k∞ ∧ kD(F −1 )k∞ , resp.
with some µ ≥ µF ∧ µF −1 .
ii) If kDF k∞ = 1 then T has slow dissipation time, τ∗ & −(α∧1) . If the noise kernel satisfies
the condition (??) for α ∈ (0, 1], then the dissipation time is simple, τ∗ ∼ −α .
iii) If µF = 1 and kDF n k∞ grows as a power-law as in Eq. (??) with β > 0, then τ̃∗ & −(α∧1)/β .
If kDF n k∞ is uniformly bounded above by a constant, then τ̃∗ = ∞ for small enough .
Remark 4. This theorem shows that classical systems on Td (i.e. C 1 diffeomorphisms) cannot have
a dissipation time growing slower than C ln(−1 ). In view of the results for toral automorphisms (cf.
Proposition 4), this lower bound on the dissipation time is sharp and consistent with Kouchnirenko’s
upper bound on the entropy of the classical systems, namely all classical systems have a finite
(possibly zero) Kolmogorov-Sinai entropy (Theorem 12.35. in [?], see also [?], [?]).
Proof of the Theorem. The following trivial lemma (similar with Lemma ??) will be crucial
in the proof.
Lemma 3. Assume that there exists some α > 0 and a strictly increasing function γ(n), γ(0) = 0
such that
(32)
∀n ≥ 1,
kTn k ≥ 1 − α γ(n).
Then the dissipation time is bounded from below as:
−1
(−1) 1 − e
τ∗ ≤ γ
(33)
α
where γ (−1) is the inverse function of γ.
The same statement holds for the coarse-grained version.
(n)
Our task is therefore to bound kTn k (resp. kT̃ k) from below. A simple computation shows
that for any f ∈ C 0 (Td ), kG f k∞ ≤ kf k∞ . Since convolution commutes with differentiation, for
f ∈ C 1 we also have k∇(G f )k∞ ≤ k∇f k∞ . We use this fact to estimate the gradient of T f :
k∇(T f )k∞ = k∇(G UF f )k∞
≤ k∇(f ◦ F )k∞ = k(∇f ) ◦ F · DF k∞
≤ k(∇f ) ◦ F k∞ kDF k∞ = k∇f k∞ kDF k∞ .
Repeating the above procedure m times, we get
(34)
k∇(Tm f )k∞ ≤ k∇f k∞ kDF km
∞,
k∇(UF Tm f )k∞ ≤ k∇f k∞ kDF km+1
∞ .
We now choose some arbitrary f ∈ C01 (Td ), with kf k = 1. We first apply the triangle inequality:
kTn f k = kG UF Tn−1 f k ≥ kUF Tn−1 f k − k(G − I)UF Tn−1 f k.
To estimate the second term on the RHS we use the bound (??) and the estimate (??) to obtain
kTn f k ≥ kTn−1 f k − Cα∧1 k∇f k∞ kDF kn∞ .
Applying the same procedure iteratively to the first term on the RHS, we finally get (remember
kf k = 1):
n
X
n
n
α∧1
kT k ≥ kT f k ≥ 1 − C k∇f k∞
(35)
kDF km
∞.
m=1
The computations in the case of the coarse-grained operator are even simpler:
kT̃(n) f k = kG UFn G f k
≥ 1 − Cα∧1 k∇f k∞ − Cα∧1 k∇(G f )k∞ kDF n k∞
(36)
≥ 1 − 2Cα∧1 k∇f k∞ kDF n k∞ .
12
Notice that from the assumptions on f , k∇f k∞ cannot be made arbitrary small, but is necessarily
larger than some positive constant. We choose some arbitrary function, say f = ek with k = (1, 0)
which satisfies k∇f k∞ = 2π.
The estimate (??) has the form given in Lemma ??. The growth of the function γ(n) depends on
whether kDF k∞ is equal to or larger than 1, which explains why the lower bounds are qualitatively
different in the two cases.
In case kDF k∞ is strictly larger than 1, then the function γ(n) grows like an exponential,
therefore the lower bound is of the type (??). For the coarse-grained version, a growth of kDF k∞
of the type (??) yields the lower bound for τ̃∗ in (??).
1−e−1 −(α∧1)
In the case kDF k∞ = 1, γ(n) is a linear function, so that τ∗ ≥ Ck∇f
.
k∞ n
In the coarse-grained version, if µF = 1 and kDF k∞ grows like in (??) with β > 0, the
dissipation is slow: τ̃∗ ≥ C−(α∧1)/β . In the case where kDF n k∞ is uniformly bounded by some
constant, the norm of the coarse-grained propagator stays larger than some positive constant for
all times, so that for small enough noise τ̃∗ is infinite.
5. An upper bound of the dissipation time for mixing maps
For any two functions f, g ∈ L20 (Td ) the dynamical correlation function for the map F is defined
as the following function of n ∈ N (see e.g. [?]):
0
Cf,g (n) = Cf,g
(n) = m(f UFn g) = hf¯, UFn gi = hPFn f¯, gi.
The same quantity may be defined for the noisy evolution:
Cf,g
(n) = m(f Tn g).
We recall that a map F is called mixing iff for any f, g ∈ L20 ,
Cf,g (n) → 0,
as
n → ∞.
The correlation function can easily be measured in (numerical or real-life) experiments, so it often
used to characterize the dynamics of a system.
To focus the attention, we will only be concerned with maps for which correlations decays in
a precise way. We assume that there exists Hölder exponents s∗ , s ∈ R+ , 0 ≤ s∗ ≤ s together
n→∞
with some decreasing function Γ(n) = Γs∗ ,s (n) with Γ(n)−−−→0, such that for any observables
s∗
f ∈ C0 (Td ), g ∈ C0s (Td ) and for sufficiently small ≥ 0 (sometimes only for = 0),
(37)
∀n ∈ N,
|Cf,g
(n)| ≤ kf kC s∗ kgkC s Γ(n).
In general, such a bound can be proven only if the map F has regularity C s+1 . The reason why we
do not necessarily take the same norm for the functions f and g will be clear below.
We will be mainly interested in the following two types of decay
i) Power-law decay: there exists C > 0, β > 0 such that,
(38)
Γ(n) = Cn−β .
This behaviour is characteristic of intermittent maps, e.g. maps possessing one or several
neutral orbits [?].
ii) Exponential decay: there exists C > 0, 0 < σ < 1 such that,
(39)
Γ(n) = Cσ n .
Such a behaviour was proved in the case of uniformly expanding or hyperbolic maps on the
torus (see Section ??), as well as many other cases [?].
13
The central result of this section is a relationship between, on one side, the decay of noisy (resp.
noiseless) correlations and on the other side, the small-noise behaviour of the noisy (resp. coarsegraining) dissipation time. The intuitive picture is similar to the one linking the local expansion
rate to the dissipation: namely, a fast decay of correlations is generally due to the transition of
“soft” into “hard” fluctuations of the observable through the evolution, which is itself induced
by large expansion rates of the map. Still, as opposed to what we obtained in last Section, the
following theorem and its corollary yields upper bounds for the dissipation time.
Theorem 3. Let F be a volume preserving map on Td with correlations decaying as in Eq. (??)
for some indices s s∗ and decreasing function Γ(n), at least in the noiseless limit = 0. Assume
that the noise generating function g is ([s] + 1)-differentiable, and that all its derivatives of order
|α|1 ≤ [s] + 1 satisfy
|Dα g(x)| .
1
,
|x|M
|x| 1,
with a power M > d.
Then there exist constants C̃ > 0, o > 0 such that the coarse-grained propagator satisfies
(40)
∀ ≤ o ,
∀n ≥ 0,
Γ(n)
.
d+s+s∗
kT̃(n) k ≤ C̃
If the decay of correlations (??) also holds for sufficiently small > 0 (and assuming the PerronFrobenius operator PF is bounded in C s (Td )), then the noisy operator satisfies (for some constants
C > 0, o > 0):
(41)
∀ ≤ o ,
∀n ≥ 0,
kTn k ≤ C
Γ(n)
d+s+s∗ )
.
From these estimates, we straightforwardly obtain the following bounds on both dissipation times
(the assumptions on F and the noise generating function g are the same as in the Theorem):
Corollary 4. I) If the correlation function satisfies the bound (??) for = 0, then the coarsegrained dissipation time is well defined (τ̃∗ < ∞). Moreover,
i) if Γ(n) ∼ n−β then there exists a constant C̃ > 0 such that
τ̃∗ ≤ C̃
∗
− d+s+s
β
ii) if Γ(n) ∼ σ n then there exists a constant c̃ such that
τ̃∗ ≤
d + s + s∗
ln(−1 ) + c̃,
| ln σ|
II) If Eq. (??) holds for sufficiently small > 0, then
i) if Γ(n) ∼ n−β , there exists a constant C > 0 such that
τ∗ ≤ C
∗
− d+s+s
β
ii) if Γ(n) ∼ σ n , there exists a constant c such that
τ∗ ≤
d + s + s∗
ln(−1 ) + c.
| ln σ|
Proof of the Theorem.
(n)
1st step: We represent the action of Tn (resp. T̃ ) on an observable f ∈ L20 (Td ) in terms of
the correlation functions C (n) (resp. C(n)). To do this we Fourier decompose both Tn+2 f and
14
f1 = UF f , and use Eq. (??):
Tn+2 f
=
X
hej , G UF Tn G f1 iej
06=j∈Zd
=
X
X
fˆ1 (k)hG ej , UF Tn G ek iej
X
X
fˆ1 (k)ĝ (j)ĝ (k)hPF ej , Tn ek iej .
06=j∈Zd 06=k∈Zn
=
06=j∈Zd 06=k∈Zd
(remember that ĝ is a real function). A similar computation for the coarse-grained propagator
yields:
X
X
fˆ(k)ĝ (j)ĝ (k)hej , UFn ek iej .
T̃(n) f =
06=j∈Zd 06=k∈Zd
Taking the norms on both sides, we get in the noisy case:
2
X X
n+2
2
n
ˆ
kT f k =
f1 (k)hPF ej , T ek iĝ (j)ĝ (k)
06=j∈Z2 06=k∈Zd
≤
X X
06=j∈Zd
(42)
=⇒ kTn+2 f k2 ≤ kf1 k2
|fˆ1 (k)|2
06=k∈Zd
X
X
|hPF ej , Tn ek i|2 |ĝ (j)ĝ (k)|2
06=k∈Zd
|CP F e−j ,ek (n)|2 |ĝ(j)ĝ(k)|2 ,
06=j,k∈Zd
and in the coarse-graining case
(43)
kT̃(n) f k2 ≤ kf k2
X
|Ce−j ,ek (n)|2 |ĝ(j)ĝ(k)|2 .
06=j,k∈Zd
These two expressions explicitly relates the dissipation with the correlation functions.
2nd step: We now apply the estimates (??) on correlations for the observables ek , e−j , PF e−j .
In the coarse-grained case, it yields (using simple bounds of the type of Eq. (??)):
∀j, k ∈ Zd \ {0},
|Ce−j ,ek (n)| ≤ C 0 |j|s |k|s∗ Γ(n).
In the noisy case, we need to assume that the Perron-Frobenius operator PF is bounded in the
space C s (Td ). This property is in general a prerequisite in the proof of estimates of the type (??),
so this assumption is quite natural here.
∀j, k ∈ Zd \ {0},
|CP F e−j ,ek (n)| ≤ C kPF e−j kC s kek kC s∗ Γ(n)
≤ CkPF kC s |j|s |k|s∗ Γ(n).
(44)
We insert these bounds on the decay of correlations in the expressions (??-??), for instance in the
coarse-grained case we get:
2
X
(n) 2
2
−(s+s∗ )
s+s∗
2
(45)
∀n ≥ 0,
kT̃ k ≤ C Γ(n) |k|
ĝ(k)
.
06=k∈Zd
3rd step: We finally estimate compute the -dependence of the RHS of the above inequality. Up
R
to a factor −d , the sum in the brackets is a Riemann sum for the integral |ξ|s+s∗ ĝ(ξ)2 dξ < ∞.
A precise connection is given in the following lemma, proven in Appendix ??:
15
Lemma 4. Let f ∈ C 0 (Rd ), decaying at infinity as |f (x)| . |x|−M with M > d. Then the following
small- estimate holds in the limit → 0:
Z
X
(46)
fˆ(ξ)2 dξ + O(M ).
d
fˆ(k)2 =
k∈Zd
Rd
Let m ∈ N satisfy 2m ≤ s + s∗ ≤ 2m + 2 (notice that m ≤ [s] since we assumed s∗ ≤ s). From
the obvious inequality
∀x > 0,
xs+s∗ ≤ x2m + x2m+2 ,
we may replace in the RHS of (??) the factors |k|s+s∗ by |k|2m + |k|2m+2 . Applying Lemma ??
to the derivatives of g of order m and m + 1, we end up with the following upper bound, which
proves the first part of the theorem:
2
Z
1
(n) 2
2
2m
2(m+1)
2
M
kT̃ k ≤ C Γ(n)
|ξ| + |ξ|
ĝ(ξ) dξ + O( )
d+s+s∗ Rd
Γ(n)2
≤ C 0 2(d+s+s ) kgk4H m+1 .
∗
The computations follow identically for the case of the noisy operator, yielding the second part of
the theorem.
6. Some examples of mixing maps
In this section we apply the results of the last two sections to several classes of measure-preserving
maps which have been proven to be mixing, with various types of correlation decays.
6.1. Noiseless correlations. Let us first consider the correlation functions Cf,g (n) for the noiseless dynamics. A common route to prove that a map F is mixing consists in studying the PerronFrobenius operator PF on some cleverly selected space B of densities (this may be a Banach or
Fréchet space). The objective is to prove that the spectrum of PF on B is of quasicompact type.
More precisely, one shows that PF admits 1 as simple eigenvalue (with the constant eigenfunction),
and that the radius of its essential spectrum is smaller than some 0 < r1 < 1. The spectrum outside the disk of radius r1 is made of a finite number of isolated, finitely-degenerate eigenvalues {λi }
of moduli r1 < |λi | < 1 (usually called Ruelle-Pollicott resonances). We order these eigenvalues
according to their decreasing moduli, so that the largest modulus is |λ1 |. This spectral structure
of PF on B implies that for any 1 > σ > max(|λ1 |, r1 ) (if the largest resonance λ1 is semisimple,
one may take σ = |λ1 |), there is some constant C = C(σ) such that for any f ∈ B0 , g ∈ B0∗ :
(47)
|Cf,g (n)| = |hg, PFn f iB∗ ,B |
≤ kgkB∗ kPFn kB0 kf kB
≤ C kgkB∗ kf kB σ n .
The choice of the space B is crucial here: in general the spectrum of PF on L20 intersects the
unit circle, so there is no chance to prove exponential decay within the L2 framework. B can be a
Banach space of bounded variation, Hölder or C s functions which embeds continuously in L2 (this
is the choice made for F uniformly expanding, see Section ??); it may also be a space of generalized
functions lying outside L2 (case of Anosov maps, see Section ??).
In general, there exist Hölder exponents 0 ≤ s∗ ≤ s such that C s (resp. C s∗ ) embeds continuously
in B (resp. its dual B ∗ ). As a result, the upper bound (??) induces that for any f ∈ C0s , g ∈ C0s∗
and any n > 0,
(48)
|Cf,g (n)| ≤ C kgkC s∗ kf kC s σ n .
This is the form of upper bound we used in Theorem ??. When the radius σ is given by a resonance
λ1 , it can often not be decreased when one takes observables of higher regularity. In this case, it is
16
preferable to take in the above estimate the weakest norms possible, that is take s and s∗ as small
as possible (to obtain better upper bounds in Corollary ??).
This strategy of proof has been applied to several types of maps [?, ?], including the (noninvertible) expanding maps and the Anosov or Axiom-A diffeomorphisms on a compact manifold. We
will give some details and examples of these two types of maps in the next subsections. Exponential
decay of correlations has also been proven (using various methods) for piecewise expanding maps
on the interval, some nonuniformly hyperbolic/expanding maps, like “good” logistic maps on the
interval, or “good” Hénon maps; some expanding or hyperbolic maps with singularities. In those
cases, the decay rate may have no (obvious) spectral interpretation, as opposed to the formalism
described above [?].
ξ
Different types of decay, like the stretched exponential Cf,g (n) ≤ Krn with 0 < r < 1 and 0 <
ξ < 1 have been proved for some Poincaré maps of hyperbolic flows and some random nonuniformly
hyperbolic systems; yet, it seems that the bound may not be optimal in some of these cases, but
rather due to the method of the proof. On the opposite, the polynomial decay Cf,g (n) . n−β
was shown to be optimal for some “intermittent” systems, like a one-dimensional map expanding
everywhere except at a fixed “neutral” point (such maps are sometimes called “almost expanding”
or “almost hyperbolic”).
Remark: In general, an expanding or Anosov map does not preserve the Lebesgue measure, so
the first step before dealing with correlations is to precise the invariant measure with respect to
which one wants to study the ergodic properties. In the “nice” cases, one can prove the existence
and uniqueness of a “physical measure”, which is ergodic for the map F , and then study the
correlation functions with respect to this measure. The formalism of Perron-Frobenius operators
applies also to this general case, the physical measure being related with the simple eigenstate of
PF (on the space B) associated with the eigenvalue 1. In the present paper we only consider maps
for which the physical measure is the Lebesgue measure.
6.2. Noisy correlations. There exist fewer results on the decay of correlations for stochastic
perturbations of deterministic maps, like our noisy evolution T . In general, one wants to prove
some sort of strong stochastic stability, that is stability of the invariant measure, and of the rate of
decay of the correlations in the limit when the noise parameter vanishes.
In the case of exponential mixing, this strong stochastic stability implies that for small enough ,
the upper bound (??) is “stable” when switching on the noise: for small enough > 0 there exists
→0
a radius σ −−→σ such that for any f ∈ C0s , g ∈ C0s∗ ,
(49)
∀n > 0,
|Cf,g
(n)| ≤ C kgkC s∗ kf kC s σn .
This stability has been proven for smooth uniformly expanding maps [?], some nonuniformly
expanding or piecewise expanding maps (see the review in [?] or the book [?]). It has been shown
also for uniformly hyperbolic (Anosov) maps on the 2-dimensional torus [?]. The proof uses perturbation theory: one shows that the isolated eigenvalues of G ◦ PF on B of moduli |λi, | > r1
(and the associated eigenstates) vary continuously w.r.to when → 0. Therefore, one can choose
a rate σ > max(|λ1, |, r1 ) which is a continuous function of .
In the next two sections, we describe in some detail the exponential decay of correlations for
smooth uniformly expanding maps and Anosov diffeomorphisms on the torus.
6.3. Smooth uniformly expanding maps. Let F be a C s+1 map on Td (with s ≥ 0). Assume
that there exists λ > 1 such that for any x ∈ Td and any v in the tangent space Tx Td , kDF (x)vk ≥
λkvk (we assume that λ is the largest such constant). Such a map is called uniformly expanding.
In general, it admits a unique absolutely continuous invariant probability measure; here we will
restrict ourselves to maps for which this measure is the Lebesgue measure.
Ruelle [?] proved that the Perron-Frobenius operator PF of such a map is quasicompact on
the space C s (Td ), and that its essential radius is contained inside the disk of radius r1 = λ−s .
17
In general, one has little information on the possible discrete spectrum outside this disk (upper
bounds on the decay rate have been obtained in the case of an expanding map of regularity C 1+η
[?]). Strong stochastic stability for such maps was proven in [?], with a more general definition of
the noise than the one we gave.
For all these cases, one can take s∗ = 0, since the continuous functions are continuously embedded
in any space (C s )∗ .
Case of a linear expanding map. We describe the simplest example possible for such a map,
namely the angle-doubling map on T1 defined as F (x) = 2x mod 1. This map is real analytic, with
uniform expansion rate λ = 2. Due to its linearity, the dynamics of this map (as well as its noisy
version) is simple to express in the basis of Fourier modes ek (x) = e2iπkx :
∀k ∈ Z, UF ek = e2k
=⇒ T ek = ĝ(k)e2k
n
Y
n
ĝ(2j k) e2n k
=⇒ T ek =
j=1
The computation is even simpler for the coarse-grained propagator:
T̃(n) ek = ĝ(k)ĝ(2n k)e2n k .
α
To fix the ideas, we take for the noise generating function ĝ(ξ) = e−|ξ| for some 0 < α ≤ 2. One
easily checks that for any n ≥ 1,
2nα − 1
kTn k = kTn e1 k = exp − α
,
1 − 2−α
kT̃(n) k = exp{−α (2nα + 1)}.
For any > 0, these decays are super-exponential: the spectrum of T on L20 is reduced to {0} for
any > 0 (the spectrum of UF is the full unit disk). From there, we get explicit expressions for
both dissipation times:
1
1
τ∗ =
(50)
ln(−1 ) + O(1),
τ̃∗ =
ln(−1 ) + O(1).
ln 2
ln 2
For this linear map, 2 = kDF k∞ = µF , so this estimate is in agreement with the lower bounds (??),
the latter being sharp if α ∈ [1, 2]. On the other hand, ln 2 is also equal with the Kolmogorov-Sinai
(K-S) entropy h(F ) of F . Therefore, for this linear map the dissipation rate constant exists, and
1
is equal to h(F
).
To compare these exact asymptotics with the upper bounds of Corollary ??, we estimate the
correlation functions Cf,g (n) on the spaces C s (T1 ). We give below a short proof in the case s > 21 .
We will use the following Fourier estimates [?]:
∃C > 0,
, ∀f ∈ C0s (T1 ),
∀k 6= 0,
|fˆ(k)| ≤ C
kf kC s
.
|k|s
Therefore, writing the correlation function as a Fourier series, we get:
X
X kf kC s 2
n
2
n
2
ˆ
kPF f k =
|f (2 k)| ≤
C n s
|2 k|
06=k∈Z
(51)
=⇒ kPFn f k ≤ C 0
06=k∈Z
kf kC s
.
(2s )n
This estimate yields a decay of the correlation function as in Eq. (??), with a rate σ = 2−s and
s∗ = 0. One can check that this rate is sharp for functions in C s : indeed, any z ∈ C, |z| < 2−s
18
is an eigenvalue of PF on that space. Applying the Corollary ??,I ii), we get that for any s ≥ 0,
there exists a constant c such that
1+s
(52)
τ̃∗ ≤
ln(−1 ) + c̃
s ln 2
for sufficiently small . The exact dissipation rate constant 1/ ln 2 is recovered only for large s.
A straightforward computation shows that the estimate (??) also holds if one replaces PF by
PF ◦ G ; hence the noisy correlation function dynamics satisfies the same uniform upper bound
as the noiseless one, with the decay rate σ = 2−s . As a result, the upper bound on τ∗ given by
Corollary ??,II ii) is the same as in Eq. (??).
6.4. Anosov diffeomorphisms on the torus. We recall that a diffeomorphism F : Td 7→ Td is
called Anosov if it is uniformly hyperbolic: there exist constants A > 0 and 0 < λs < 1 < λu such
that at each x ∈ Td the tangent space Tx Td admits the direct sum decomposition Tx Td = Exs ⊕ Exu
into stable and unstable subspaces such that for every n ∈ N,
(Dx F )(Exs ) = EFs x ,
(Dx F )(Exu ) = EFu x ,
k(Dx F n )|Exs k ≤ Aλns ;
k(Dx F −n )|Exu k ≤ Aλ−n
u .
These inequalities have obvious consequences on the expansion rates of F and F −1 , for instance they
imply kDF kn∞ ≥ kDF n k∞ ≥ A−1 λnu . As a consequence, the quantities of interest in Theorem ??,
i) satisfy
kDF k∞ ∧ kDF −1 k∞ ≥ λu ∧ λ−1
s ,
µF ∧ µF −1
≥ λu ∧ λ−1
s .
All these expansion rates are > 1, so both noisy and coarse-grained dissipation times admit logarithmic lower bounds as in Eq. (??).
To obtain upper bounds, we use the mixing properties of this dynamics. Exponential mixing
has been proven for Anosov diffeomorphisms of regularity C 1+η (0 < η < 1) by Bowen [?], using
0
symbolic dynamics; the exponential decay is then valid for Hölder observables in C η for some
0 < η 0 < η.
Because we are interested in the noisy dynamics as well, we also refer to a more recent work [?]
concerning C 3 Anosov maps on Td , which bypasses symbolic dynamics. The authors construct an
ad hoc invariant Banach space B of generalized functions on the phase space, such that the PerronFrobenius operator is quasicompact on this space. The difficulty compared to the case of expanding
maps, is that the space B explicitly depends on the (un)stable foliations generated by the map F
on Td . Vaguely speaking, the elements of B are “smooth” along this unstable direction Exu , but
can be singular (“dual of smooth”) along the stable foliation Exs . The space B is the completion
of C 1 (Td ) with respect to a weaker norm k · kB adapted to these foliations. The definition of that
norm is given in terms of a parameter 0 < β < 1, the choice of which depends on the regularity
of the unstable foliation. In general, the latter is only Hölder continuous on the torus, with some
exponent 0 < τ < 2 (note that the regularity of the foliations has little to do with the regularity of
the map itself: a real-analytic map may have foliations being only Hölder). Then, one must take
β < τ ∧ 1, and the authors prove that the essential spectrum of PF on B has a radius smaller than
β
r1 = max(λ−1
u , λs ). This upper bound is sharper if β can be taken close to 1, that is, if the foliation
is at least C 1 . This is the case for smooth area-preserving Anosov maps on T2 , for which the
foliations have regularity C 2−δ for any δ > 0 [?]. The operator PF may have isolated eigenvalues
1 > |λi | > r1 , corresponding to eigenstates in B which are genuine distributions, outside L20 . As
opposed to the radius r1 , there is (to our knowledge) no simple general upper bound for largest
resonance |λ1 | in terms of (λu , λs ).
The space C 1 (Td ) can be embedded continuously in both B and its dual B ∗ , so that one can take
s = s∗ = 1 in Eq. (??). Therefore, for any 1 > σ > max(|λ1 |, r1 ), there is some constant C > 0
19
such that for any f , g ∈ C01 (Td ),
(53)
∀n > 0,
|Cf,g (n)| ≤ C kgkC 1 kf kC 1 σ n .
In the proof of Theorem ?? (Step 3), for the case s = s∗ = 1 we only need to assume that the
noise generating function g is C 1 with fast-decaying first derivatives. The fast decay implies that
the first moment of g is finite (that is, one can take α ≥ 1).
The results of [?] also concern the noisy dynamics. If the unstable foliation has regularity
C 1+η with η > 0 (for instance for any C 3 Anosov diffeomorphism on the 2-dimensional torus),
and assuming the noise generating function g ∈ C 2 (Rd ) with compact support, the authors prove
the strong spectral stability of the Perron-Frobenius operator PF on the space B defined with the
(n)
parameter β < η. Therefore, the estimate (??) also applies to the noisy correlation function Cf,g
as long as is small enough.
We collect below the results regarding the dissipation time of C 3 Anosov maps on the torus.
Theorem 4. Let F be a volume preserving C 3 Anosov diffeomorphisms on Td and let g be a C 1
noise generating function with fast decay at infinity.
I) Then there exist µ ≥ λu ∧ λ−1
s , 0 < σ < 1 and C̃ > 0 such that the dissipation time of the
coarse-grained dynamics satisfies
1
d+2
ln(−1 ) − C̃ ≤ τ̃∗ ≤
ln(−1 ) + C̃,
ln µ
| ln σ|
II) If in addition F has C 1+η -regular foliations and g ∈ C 2 (Rd ) is compactly supported, then the
dissipation time of the noisy evolution satisfies for some C > 0 and small enough :
1
d+2
ln(−1 ) − C ≤ τ∗ ≤
ln(−1 ) + C
ln kDF k∞
| ln σ|
6.5. Ergodic linear automorphisms of the torus. In this section we describe examples of
Anosov maps for which the dissipation time can be exactly determined. One can even determine
the value of the dissipation rate constant; we will mention the possible connection of the latter
with the Kolmogorov-Sinai entropy. After the simple example of a (generalized) cat map on the 2dimensional torus [?], we recall the results obtained in [?] for d-dimensional ergodic automorphisms.
Throughout this section, we take a noise generating function of the type (??) for a certain
α ∈ (0, 2], with Q = I:
X
α
e−|k| ek (x).
g,α (x) :=
k∈Zd
Notice that this function is not compactly supported, that is it does not satisfy stricto sensu the
assumptions required in [?] to prove the strong spectral stability of PF .
2-dimensional cat map. We take A ∈ SL(2, Z) with |T rA| > 2 and define the dynamics as F (x) =
FA (x) = At x mod 1 (At is the transposed matrix of A). This map is of Anosov type, and conserves
the area by definition. The dynamics is easy to express on the Fourier modes:
UF ek = eAk
Tn ek = e−
Pn
l=1
|Al k|α
=⇒ kTn k = exp −α
eAn k
!
n
X
min
|Al k|α .
06=k∈Z2
l=1
Similarly,
kT̃(n) k
α
= exp −
min (|k| + |A k| ) .
α
06=k∈Z2
20
n
α
Let us call λ, λ−1 the eigenvalues of A, with the convention |λ| > 1. One can easily show that there
are two constants 0 < C1 < C2 such that for any n > 0,
C1 |λ|nα/2 ≤
nα/2
C1 |λ|
≤
min
06=k∈Z2
n
X
|Al k|α ≤ C2 |λ|nα/2 ,
l=1
min (|k|α + |An k|α ) ≤ C2 |λ|nα/2 .
06=k∈Z2
The above estimate yields the following asymptotics in the small- limit:
τ∗ =
2
ln(−1 ) + O(1),
ln |λ|
τ̃∗ =
2
ln(−1 ) + O(1).
ln |λ|
As in the case of linear expanding map, the dissipation rate constant exists, and from the identity
h(F ) = ln |λ|, can be related with the K-S entropy of the linear map F .
Let us compare these exact asymptotics with the bounds obtained in the previous sections.
The cat map being linear, one has kDF k∞ = kAt k ≥ |λ|, kDF −1 k∞ = k(At )−1 k ≥ |λ| (with
equality iff A is symmetric). Since A is diagonalizable, we have kDF n k∞ = k(At )n k ∼ |λ|n , so that
µF = µF −1 = |λ|. Therefore, the lower bounds for the dissipation times given in Theorem ??i) are
strictly smaller then the exact rates derived above, they differ from the latter by a factor ≤ 1/2.
To estimate the dynamical correlations, we use Fourier decomposition and proceed as in [?]:
we construct a “primitive subset” S of the Fourier plane Z2 \ {0}, such that each Fourier orbit
intersects S once and only once. This subset looks like the union of 4 angular sectors of the plane,
and has the following crucial property: that
(54)
∃c > 0,
∀k ∈ S,
∀n ∈ Z,
|An k| ≥ c|λ||n| |k|.
The correlation function may be written as:
X
XX
fˆ(−k)ĝ(An k) =
fˆ(−Al k)ĝ(Al+n k).
Cf,g (n) =
06=k∈Z2
k∈S l∈Z
Let us assume s > 1, and take f ∈ C0s (T2 ): its Fourier coefficients decrease for all 0 6= k ∈ Z2 as
|fˆ(k)| ≤ C
kf kC s
.
|k|s
Using this upper bound as well as the estimate (??), we find that
XX
1
|Cf,g (n)| ≤ C 2 kf kC s kgkC s
.
|l+n|+|l|
2
2 )s
(c
|λ|
|k|
k∈S l∈Z
The sum over k converges, and the sum over l is bounded from above by |λ|Cns . This implies that for
any s > 1, the correlation function decreases as in Eq. (??), with the rate σ = |λ|−s . Actually, this
decrease holds as well in the case 0 < s ≤ 1, but the proof is different. From there, Corollary ??I
ii) yields the upper bound
(55)
τ̃∗ ≤
2 + 2s
ln(−1 ) + c.
s ln |λ|
This method can be straightforwardly adapted to prove that the noisy correlation function
decreases as fast as the noiseless one (in case s > 1). As a result, we obtain the same upper
bound for τ∗ as for τ̃∗ . We notice that, as in the case of the angle-doubling map, the constant in the
upper bound converges to the exact rate only in the limit s 1, while for s = 1 (cf. the discussion
preceding Thm. ??), the constant is twice larger.
(n)
Cf,g
21
Toral automorphisms in higher dimensions. We consider toral automorphisms F = FA given by
matrices A ∈ SL(d, Z); such an automorphism is ergodic iff none of the eigenvalues of A are roots
of unity. The K-S entropy h(F ) of F is given by the formula [?]
X
(56)
h(F ) =
ln |λj |,
|λj |>1
where λj are the eigenvalues of F . All ergodic toral automorphisms have positive entropy. We
denote by P the characteristic polynomial of the matrix A and by {P1 , ..., Pr } the complete set of
its distinct irreducible factors (over Q). Let dj denote the degree of the polynomial Pj and hj the
K-S entropy of a toral automorphism with the characteristic polynomial Pj . For each Pj we define
its dimensionally averaged K-S entropy as
hj
ĥj = .
(57)
dj
For the whole matrix F we define its minimal dimensionally averaged entropy (denoted ĥ(F )) as
ĥ(F ) = min ĥj
j=1,...,r
Notice that for an irreducible matrix A, that is a matrix admitting no proper invariant rational
)
subspace (see [?]), this quantity reduces to h(F
d .
Our notation regarding the noise parameter slightly differs from the one used in [?] (one has to
replace ε2α by ); the results obtained there read as follows:
Proposition 4. [?] Let F = FA be a toral automorphism on Td , and assume the noise kernel to
be as in Eq. (??). Then, in the limit of small noise,
i) both dissipation times have a power-law behaviour iff F is not ergodic.
ii) both dissipation times have logarithmic behaviour iff F is ergodic.
iii) if F is ergodic and A is diagonalizable then
1
τ∗ ≈ τ̃∗ ≈
ln(−1 ).
ĥ(F )
We end this section with a remark that the dissipation time is independent of the subtleties
of the asymptotics of the corresponding correlation function. Indeed, we illustrate this fact by
restating a well know result regarding super-exponential decay of correlations for d-dimensional
toral automorphisms as a corollary of the above proposition.
Corollary 5. Let F be a diagonalizable ergodic toral automorphism and λ any constant such that
0 < λ < ĥ(F ). Then for any f, g ∈ L20 (T2d )
2 λn
Cf,g
(n) ≤ kf kkgke−
Let f, g ∈ G (L20 (T2d )) be smooth observables than
2 λn
−1
−
Cf,g (n) ≤ kG−1
f kkG gke
.
7. Conclusion
We have investigated the effect of noise or coarse-graining on the dynamics generated by a conservative map, in particular the connection between the speed of dissipation of the noisy dynamics
and the spectral and dynamical properties of the underlying map.
We restricted ourselves to the case of volume preserving maps on the d-dimensional torus. The
choice of the torus allowed us to use the Fourier transformation in our proofs, that harmonic analysis
on that manifold. Most of our results can certainly be generalized to volume preserving maps on
other compact Riemannian manifolds. The choice of the torus was also guided by the existence
of simple volume preserving Anosov maps on it, most notably the linear examples presented in
22
Section ??. As explained at the end of Section ??, one may also want to extend the results to
maps which do not leave invariant the Lebesgue measure, but still admit a ‘physical measure’, as
is the case for uniformly expanding or Anosov maps (the physical measure is of SRB type, that is,
its projection along the unstable manifold is absolutely continuous [?]). The noisy perturbations
of these maps have been analyzed as well [?, ?], in particular the strong spectral stability of the
Perron-Frobenius operator holds under the same conditions as for the volume preserving maps.
Although the equilibrium measure is more complicated, it might be possible to define and study a
dissipation time in this more general framework.
As we explained in Corollary ??, the dissipation of a non-weakly-mixing map is governed only
by the nontrivial eigenstates of the Koopman operator, the speed of dissipation depending on the
smoothness of these eigenstates. The asymptotics of the noisy dissipation time is generally a power
law in −1 , while there remains a possibility of faster dissipation if all eigenstates are singular
enough (see Remarks after Corollary ??).
The more interesting results concern the mixing dynamics, in particular when the mixing occurs
exponentially fast. We then proved that both noisy and coarse-graining dissipation times behaved
as the logarithm of −1 in the small noise limit. This dependence can be understood through
the time evolution in Fourier space: a mixing map transforms long wavelength fluctuations into
short wavelength ones, the latter being damped fast by the noise operator. This Fourier evolution
is typical of uniformly expanding/hyperbolic maps; it is already responsible for the fast decay of
dynamical correlations. The link between this decay and the dissipation is explicited in Theorem ??,
and its application to Anosov maps is given in Theorem ??.
In this context, we were unable to solve the problem of the existence and value of the dissipation
rate constant (that is, the prefactor in front of ln(−1 )). We obtained lower, resp. upper bounds
for this constant, in terms of the local expanding rate, resp. the rate of decay of correlations. It is
not clear whether this constant can in general be related with the measure-theoretic (KS) entropy
of the map, as is the case for linear automorphisms (modulo some algebraic subtleties [?]).
Although we used the spectral estimates of Theorem ?? mostly for the case of non-weakly-mixing
maps, it also makes sense to use that theorem in the reverse direction in the case of mixing maps,
that is, deduce (pseudo)spectral properties of the noisy propagators, starting from the dissipation
time estimates obtained in Corollary ??. The analysis of the pseudospectrum of T for mixing maps
would complement the spectral one [?, ?]. We did not enter this aspect in the main text, because
our attention was devoted to obtaining information on the dissipation time.
The results of the present paper concern classical (i.e. non-quantum) dynamical systems. The
quantization of both linear and nonlinear maps on a symplectic (even-dimensional) torus as a phase
space has been studied in a number of works [?, ?, ?, ?, ?]. Several recent studies deal with some
form of noise, or decoherence, in discrete-time quantum dynamics [?, ?, ?, ?, ?]. While most of
these works concentrate on spectral or entropic properties of noiseless/noisy dynamics, the long
time behaviour of the quantum system can also be studied from the dissipation time point of view.
The quantum setting provides a natural framework for extension of the present work, which we
will address in a separate paper [?].
Acknowledgments: This project started during a stay of S.N. at the Mathematical Sciences
Research Institute (Berkeley), the support of which is gratefully acknowledged. The authors would
like to thank Prof. S. De Bièvre and Prof. Nachtergaele for putting them together in contact.
Appendix A. Proofs of some elementary facts
A.1. Proof of Lemma ??. We use the following upper bound: for any α ∈ (0, 2], there is a
constant Cα such that
(58)
∀x ∈ R,
0 ≤ 1 − cos(2πx) ≤ Cα |x|α .
23
2
Besides, one has the asymptotics 1 − cos(x) ≈ x2 for small x. We simply apply these estimates to
the following integral:
Z
1 − ĝ(ξ) =
(1 − cos(2πx · ξ))g(x)dx
Rd
Z
≤
Cα |x · ξ|α g(x)dx
d
R
Z
≤ Cα |ξ|α
|x|α g(x)dx = Cα Mα |ξ|α .
Rd
In the case g admits a second moment, we have in the limit ξ → 0:
Z
Z
(1 − cos(2πx · ξ))g(x)dx ≈
2π 2 (x · ξ)2 g(x)dx
Rd
Rd
Z
≈ 2π 2 |ξ|2
(x · ξ̂)2 g(x)dx,
Rd
where we have used the notation ξ̂ =
ξ
|ξ|
for any ξ 6= 0.
A.2. Proof of Proposition ?? . The statement i) is standard in the context of distributions [?,
p.157]. In our case, assume that f ∈ L2 is normalized to unity and consider an arbitrary small
P
2
ˆ
δ > 0. Since f ∈ L2 (Td ), there exists K > 0 s.t.
|k|≥K |f (k)| < δ. Since ĝ is continuous and
ĝ(0) = 1, there exists η such that (1 − ĝ(ξ))2 < δ if |ξ| < η. Thus using spectral decomposition
η
(??) of G , we obtain for all < K
X
X
X
kG f − f k2 =
(59)
(1 − ĝ(k))2 |fˆ(k)|2 ≤ δ
|fˆ(k)|2 +
|fˆ(k)|2 ≤ 2δ.
|k|<K
k∈Zd
|k|>K
To prove the next statement, first notice that if g satisfies the estimate (??) for the exponent
α, it also satisfies it for the exponent γ ∧ α. Using once again spectral decomposition of G , and
applying the estimate (??) with the latter exponent we get
X
kG f − f k2 ≤
(Cγ∧α Mγ∧α |k|γ∧α )2 |fˆ(k)|2
k∈Zd
(60)
≤ (Cγ∧α Mγ∧α )2 2(γ∧α)
≤
To obtain the last statement, we
that its gradient satisfies
X
|k|2(γ∧α) |fˆ(k)|2
k∈Zd
(Cγ∧α Mγ∧α )2 2(γ∧α) kf k2H γ∧α .
notice that any f ∈ C 1 (Td ) is
k∇f k2∞ ≥ k∇f k2 = 4π 2
X
|k|2 |fˆ(k)|2 ≥ 4π 2
k∈Zd
X
automatically in H 1 (Td ), and
|k|2(1∧α) |fˆ(k)|2 .
k∈Zd
The inequality (??) with γ = 1 then yields the desired result.
→0
A.3. Proof of Corollary ??. We prove the limit d (1)−−→0 by contradiction. Assume that there
is some constant a ∈ (0, 1) such that for all > 0, the distance d (1) > a. We will show that the
following triangle inequality holds:
(61)
∀ > 0,
d (1 − a/2) > a/2.
First of all, notice that the assumption d (1) > a means that for any λ ∈ S 1 , kR (λ)k < a−1 . We
apply the following identity [?]:
X
R (λ0 ) = R (λ){1 +
(λ − λ0 )n R (λ)n }
n≥1
24
1
with λ0 = rλ, for 1 − a < r < 1. Taking norm of both sides yields the bound kR (λ0 )k ≤ r−(1−a)
,
0
uniformly w.r.t . Since this upper bound holds for any |λ | = r, it shows that the spectral radius
rsp (T ) ≤ 1 − a, and proves (??) by taking r = 1 − a/2. We can now use (??) in the upper bound
(??) of Theorem ??: this -independent upper bound shows that τ∗ remains finite in the limit
→ 0, which contradicts Proposition ??.
A.4. Proof of Lemma ??. Considering its decay at infinity, the function f is automatically in
L2 (Rd ). The function fˆ2 is the Fourier transform of the self-convolution f ∗ f . Therefore, the
Poisson summation formula applied to the LHS of (??) reads
X
X
n
(62)
.
d
fˆ(k)2 =
(f ∗ f )
d
d
k∈Z
n∈Z
A simple computation shows that (f ∗ f )(x) also decays as fast as |x|−M . This piece of information
is now sufficient to control the RHS of (??), yielding the result, Eq. (??).
A.5. Proof of Corollary ??. It was showed in [?] that for any δ > 0 in case of Gaussian noise
2 e2(1−δ)ĥ(F )n
kTn k ≤ e−
(63)
.
Using this estimate one immediately gets
2 n
Cf,g
(n) = hf¯, Tn gi ≤ kf kkgkkTn k ≤ kf kkgkke− λ .
Now let f = G f0 and g = G g0 . Since the estimate (??) holds also in coarse-grained version, we
have
Cf,g (n) = hf¯, U n gi = hG f¯0 , U n G g0 i = hf¯0 , T̃ (n) g0 i
F
≤
F
kf0 kkg0 kkT̃(n) k
−2 λn
≤ kf0 kkg0 ke
.
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