Lecture 11
1
Brownian motion: characterization
Brownian motion is named after the botanist Robert Brown, who first observed seemingly
random motion of pollen grains suspended in fluids. A rigorous mathematical construction of
Brownian motion was first given by Norbert Wiener, and for this reason, Brownian motion is
also known as the Wiener process.
Theorem 1.1 [Characterization of Brownian motion] There exists a real-valued stochastic process (Bt )t≥0 , called the (one-dimensional) standard Brownian motion, whose law is
uniquely characterized by the following properties:
(i) B has independent increments, i.e., for any t0 < t1 < · · · < tn < ∞, Bt1 − Bt0 ,
Bt2 − Bt1 , · · · , Btn − Btn−1 are independent.
(ii) B0 = 0, and for any 0 ≤ s < t, Bt − Bs is distributed as a centered Gaussian random
variable with variance t − s.
(iii) Almost surely, Bt is continuous in t.
Because Bt is almost surely continuous in t, Brownian motion can also be regarded as a
random variable taking values in the space of continuous functions C([0, ∞), R), which we
equip with the topology of uniform convergence on compact sets, which in turn generates a
Borel σ-algebra. On C([0, T ], R) for finite T , we simply use the sup-norm topology.
One may wonder why don’t we regard Brownian motion as a random variable taking values
in the space of real-valued measurable functions defined on [0, ∞), for which the natural σalgebra is the one generated from the coordinate maps t : f → f (t) ∈ R for t ≥ 0? The reason
is that in such a σ-algebra, every measurable set is determined by the values of the functions
at at most a countable number of coordinates, and the set of continuous functions is not even
measurable!
Theorem 1.1 can be proved by first constructing Bt for t ∈ Q2 := {m2−n : m, n ≥ 0},
which is possible by Kolmogorov’s extension theorem. The hard part is then to show that
(Bt )t∈Q2 is almost surely uniformly continuous on [0, T ] ∩ Q2 for any 0 < T < ∞, which
allows the extension of Bt from t ∈ Q2 to a continuous function on [0, ∞). We will give a
proof below, which will furthermore establish that Brownian motion is almost surely Hölder
continuous with exponent γ for any γ < 1/2.
Because B has independent Gaussian increments, the finite-dimensional distributions of
B, i.e., the law of (Bt1 , · · · , Btn ) for any 0 ≤ t1 < · · · < tn < ∞, are multivariate normal distributions. This qualifies B as a Gaussian process. A continuous Gaussian process is uniquely
determined by its finite-dimensional distributions, while multivariate normal distributions are
~ := (X1 , · · · , Xd ) ∈ Rd has
uniquely determined by their covariance matrices — recall that if X
multivariate normal distribution with mean E[Xi ] = 0 and covariance matrix E[Xi Xj ] = Cij
h~
x,C −1 ~
xi
−
1
~ has distribution
2
d~x on Rd . Therefore we have
for 1 ≤ i, j ≤ d, then X
d
1 e
(2π) 2 (detC) 2
the following alternative characterization of Brownian motion.
1
Theorem 1.2 [Brownian motion as a Gaussian process] The law of a standard Brownian motion (Bt )t≥0 is uniquely determined by the following properties:
(i) (Bt )t≥0 is a Gaussian process, i.e., its finite-dimensional distributions are multi-variate
normal.
(ii) E[Bs ] = 0 and E[Bs Bt ] = s for all 0 ≤ s ≤ t < ∞.
(iii) Almost surely, Bt is continuous in t.
We now collect some properties of Brownian motion. First some invariance properties.
Theorem 1.3 [Invariance properties] Let B := (Bt )t≥0 be a standard Brownian motion.
Then B satisfies the following invariance properties:
(1) [Translation] For any t0 ≥ 0, (Bt0 +t − Bt0 )t≥0 is equally distributed with (Bt )t≥0 .
(2) [Diffusive scaling] For any a > 0, B√ata t≥0 is equally distributed with (Bt )t≥0 .
(3) [Time inversion] Xt := tB1/t with X0 := 0 is equally distributed with (Bt )t≥0 .
Proof. (1) and (2) follow easily by verifing that Properties (i)–(iii) in Theorem 1.1 are
all preserved by translation and diffusive scaling. For (3), note that (Bt )t>0 and (tB1/t )t>0
are both continuous Gaussian processes, whose laws are uniquely determined by their finite
dimensional distributions, and in turn by their covariances. Since for all 0 < s < t < ∞,
E[Bs Bt ] = s,
E[sB1/s tB1/t ] = stt−1 = s,
(tB1/t )t>0 and (Bt )t>0 must be equally distributed, and so are (tB1/t )t≥0 and (Bt )t≥0 .
2
Brownian motion: path properties
Next we list some almost sure path properties for the standard Brownian motion.
Theorem 2.1 [Path properties] Let (Bt )t≥0 be a standard Brownian motion. Then almost
surely,
(i) (Bt )t∈[0,1] is Hölder continuous with exponent γ for any γ < 1/2, i.e.,
|Bt − Bs |
< ∞.
γ
0≤s<t≤1 |t − s|
sup
(2.1)
(ii) (Bt )t∈[0,1] is not Hölder continuous with exponent γ for any γ ≥ 1/2, i.e.,
|Bt − Bs |
= ∞.
γ
0≤s<t≤1 |t − s|
sup
(2.2)
Furthermore, (Bt )t≥0 is not Hölder continuous at any point t ≥ 0 with exponent γ for
any γ > 1/2, i.e.,
lim sup
s→t
|Bs − Bt |
=∞
|s − t|γ
1
for all t ≥ 0 and γ > .
2
In particular, (Bt )t≥0 is almost surely nowhere differentiable.
2
(2.3)
(iii) [Law of the iterated logarithm]
lim sup √
t→∞
Bt
Bt
= lim sup q
= 1,
2t log log t
t→0
2t log log 1
t
Bt
Bt
lim inf √
= lim inf q
= −1.
t→∞
t→0
2t log log t
2t log log 1t
(2.4)
In particular, (Bt )t≥0 visits every point in R infinitely often.
Remark 2.2 Although (Bt )t≥0 is a.s. nowhere differentiable, its distributional derivative Ḃt
can be defined via
Z ∞
Z ∞
Ḃt f (t)dt := −
B(t)f 0 (t)dt
for all f ∈ Cc∞ ([0, ∞), R).
0
0
This random distribution Ḃt is called White Noise. The fact that (Bt )t≥0 has stationary
independent increments implies Ḃt is translation invariant and
R independent in time. More
∞
distributed
precisely,
for any f, g ∈ Cc ([0, ∞), R), the random variable f (t)Ḃt dt is equally
R
R
with
f
(t
+
x)
Ḃ
dt
for
any
x
>
0,
and
if
g
has
disjoint
support
from
f
,
then
f
(t)
Ḃt dt and
t
R
R
g(t)Ḃt dt areR independent. In fact fR(t)Ḃt dt is a Gaussian random variable with mean 0
and variance f (t)2 dt. The map f :→ f (t)Ḃt dt Ris an isometry from L2 (R) → L2 (Ω, F, P),
and hence we can construct the stochastic integral f (t)Ḃt dt for a general f ∈ L2 (R) by first
constructing it for f (t) = 1[a,b) (t) and linear combinations of such indicator functions, and
then take closure in L2 (Ω, F, P).
The key to proving the a.s. Hölder continuity of a Brownian motion is the following.
Theorem 2.3 [Kolmogorov’s Moment Criterion] Let X be a real-valued stochastic process indexed by Q2 = {m2−n : m, n ≥ 0}. If for some α, β > 0,
E|Xt − Xs |β ≤ K|t − s|1+α
∀ s, t ∈ Q2 ∩ [0, 1],
(2.5)
then for any 0 < γ < αβ , almost surely there exists C := C(ω) ∈ (0, ∞) such that
|Xq − Xr | ≤ C|q − r|γ
∀ q, r ∈ Q2 ∩ [0, 1].
(2.6)
In particular, X can be extended almost surely to a continuous process indexed by [0, 1], such
that (2.6) holds for any q, r ∈ [0, 1].
Proof. Fix 0 < γ < αβ . Let Dn := {i2−n : 0 ≤ i ≤ 2n }. For any s := i2−n < t := j2−n ∈ Dn ,
by (2.5) and Chebychev inequality,
P(|Xt − Xs | > |t − s|γ ) ≤ K|t − s|1+α−βγ = K|i − j|1+α−βγ 2−n(1+α−βγ) .
(2.7)
If we let Gn := {|Xi2−n − Xj2−n | ≤ |j − i|γ 2−nγ ∀ 0 ≤ i, j ≤ 2n , |i − j| ≤ 2nη } for some fixed
η ∈ (0, 1), then by (2.7),
X
P(Gcn ) ≤
K(j − i)1+α−βγ 2−n(1+α−βγ) ≤ K2−n ,
(2.8)
0≤i<j≤2n
j−i≤2nη
where = (1 − η)(α − βγ) − 2η can be made positive if η > 0 is sufficiently small. Fix such
an η. Then by Borel-Cantelli, almost surely there exists Nω ∈ N such that ∩n≥Nω Gn occurs.
3
On the event ∩n≥Nω Gn , we will use the triangle inequality to deduce (2.6). Note that it
suffices to show that (2.6) holds for some C(ω) for all q, r ∈ Q2 ∩[0, 1] with |q −r| ≤ 2−Nω (1−η) ,
since for general q, r ∈ Q2 ∩[0, 1], we only need to replace C(ω) by C(ω)2Nω (1−η) by the triangle
inequality. Now assume 0 ≤ r < q ≤ 1 and q − r < 2−Nω (1−η) . We can find an optimal scale
m ≥ Nω such that
2−(m+1)(1−η) ≤ q − r < 2−m(1−η) .
(2.9)
By binary expansion for q and r, we can write
q=
i2−m + 2−q1 + · · · 2−qk
r = j2−m − 2−r1 − · · · 2−rl ,
where m < q1 < · · · < qk and m < r1 < · · · < rl . Since q − r ≥ (i − j)2−m and i − j ≥ −1, by
(2.9), we have |i − j| ≤ 2mη . Since ∩n≥Nω Gn occurs and m ≥ Nω , we have
|Xi2−m − Xj2−m | ≤ 2−m(1−η)γ .
(2.10)
By the triangle inequality,
|Xq − Xi2−m | ≤
k
X
2−qσ γ ≤
X
2−σγ = Cγ 2−mγ ,
(2.11)
σ>m
σ=1
where Cγ = 2γ1−1 . Similarly, the same bound also holds for |Xr − Xj2−m |. Combining the
above estimates and apply triangle inequality once more, we get
|Xq − Xr | ≤ 2−m(1−η)γ + 2Cγ 2−mγ ≤ (1 + 2Cγ )2(1−η)γ 2−(m+1)(1−η)γ ≤ C(ω)|q − r|γ
with C(ω) = (1 + 2Cγ )2(1−η)γ , and we are done.
Remark 2.4 (Moment Criterion for Continuity and Tightness) The proof of Theorem 2.3 can be easily modified to show that given a family of random continuous functions
(n)
{(Xt )t∈[0,1] }n∈N , if (2.5) holds for some K, α, β uniformly in n ∈ N, then for any > 0,
we can construct a compact set of Hölder continuous functions H ⊂ C([0, 1], R), such that
P(X (n) ∈ H ) ≥ 1 − for all n ∈ N. This implies tightness for the family of C([0, 1], R)-valued
random variables {X (n) }n∈N . There is also an analogue of (2.5) in dimensions d ≥ 1 with 1+α
in (2.5) replaced by d + α, when (Xt )t∈[0,1] is Rd -valued, see e.g. Kallenberg [K97, Corollary
14.9]. Garsia, Rodemich and Rumsey have given more refined continuity criterion in [GRR70],
which was later extended to dimensions d ≥ 1 by Garsia in [G72].
Proof of (2.1). Note that for any 0 ≤ s < t ≤ 1, Bt − Bs is a centered Gaussian random
variable with variance t − s. Therefore
Z
β
β
x2
1
β
β
2
2
E|Bt − Bs | = |t − s| E|B1 | = f (β)|t − s| ,
where f (β) = √
|x|β e− 2 dx < ∞.
2π R
Therefore (2.5) holds with α = β2 − 1 for any β > 2. Since αβ → 12 as β ↑ ∞, by Theorem 2.3,
(Bt )0≤t≤1 is a.s. Hölder continuous with exponent γ for all γ ∈ (0, 21 ).
Proof of (2.2). It suffices to verify (2.2) for γ = 12 . Note that
K(ω) :=
n
o
|Bt − Bs |
|Bt − Bs |
√
√
≥ max
sup
:1≤i≤n ,
i−1
t−s
t−s
0≤s<t≤1
≤s<t≤ i
sup
n
4
n
(i)
where Kn (ω) := sup i−1 ≤s<t≤ i
n
n
|Bt −Bs |
√
,
t−s
1 ≤ i ≤ n, are independent because (Bt )t≥0 has
(i)
independent increments, and furthermore, Kn are equally distributed with K because
√
n(Bi/n+t/n − Bi/n ),
0 ≤ t ≤ 1,
is equally distributed with (Bt )0≤t≤1 . Therefore for any M > 0,
P(K ≤ M ) ≤ P(Kn(1) ≤ M, · · · , Kn(n) ≤ M ) = P(K ≤ M )n
for all n ∈ N.
Since P(K > M ) > 0 for all M > 0, we must have P(K ≤ M ) = 0 for all M > 0. Therefore
K = ∞ a.s.
Proof of (2.3). The
and the observation that Bt
√ argument is based on time discretization
1
is typically of order t. In particular, for any γ > 2 , by the density of Gaussian distribution,
1
P(|Bt | < Ctγ ) = P(|B1 | < Ct
γ− 12
Z
)=
Ctγ− 2
1
−Ctγ− 2
x2
1
1
2C
√ e− 2 dx ∼ √ tγ− 2
2π
2π
as t ↓ 0. (2.12)
It suffices to consider only (Bt )t∈[0,1] . Fix γ > 12 , and fix k ∈ N whose exact value will only
depend on γ and will be determined later. Define
An :=
∃ t ∈ [0, 1] s.t. |Bt+h − Bt | ≤ |h|γ ∀ |h| ≤
2k .
n
Note that An is an increasing sequence of events, and
n
o
[
|Bs − Bt |
∃ t ∈ [0, 1] s.t. lim sup
<
1
⊂
An .
|s − t|γ
s→t
n∈N
m+1
γ
On the event An , let t ∈ [ m
n , n ) for some 0 ≤ m ≤ n such that |Bt+h − Bt | ≤ |h| for all
|h| ≤ 2k
n . By the triangle inequality, we have the occurrence of the event
n
k γ
Fm := B j+1 − B j ≤ 2
n
n
n
o
∀ j ∈ {m, m + 1, · · · , m + k − 1} .
Therefore by the independent increments properties of (Bt )t≥0 ,
P(An ) ≤
n
X
k
1
P(Fm ) = (n + 1)P(F0 ) = (n + 1)P |B1 | ≤ 2k γ n 2 −γ ,
m=0
1
which by (2.12) is asymptotically of the order Ck n1−k(γ− 2 ) for some Ck depending only on k,
and it tends to 0 as n → ∞ if k > (γ− 21 )−1 . For such a choice of k, we have limn→∞ P(An ) = 0.
Since the sequence of events An increases in n, we have P(An ) = 0 for all n ∈ N. In particular,
we have shown that almost surely,
lim sup
s→t
Since this is true for all γ >
implies (2.3).
1
2,
|Bt − Bs |
≥1
|t − s|γ
for all t ∈ [0, 1].
by slightly adjusting the values of γ, the above inequality
Proof of (2.4). By symmetry, it suffices to prove the limsup statement in (2.4). We will
prove the case t → ∞, the statement for t → 0 then follows from the fact that (tB1/t )t>0 is
equally distributed with (Bt )t>0 .
5
First we recall that by the reflection principle and diffusive scaling invariance of the standard Brownian motion,
r Z ∞
√
x2
2
P0 ( sup Bs > a) = 2P0 (Bt > a) = 2P0 (B1 > a/ t) =
e− 2 dx.
(2.13)
π √a
0≤s≤t
t
Note that
Z ∞
y2
x2
e− 2 dx ≤ y −1 e− 2 for all y > 0
Z
∞
and
e−
x2
2
dx ∼ y −1 e−
y2
2
as y → ∞.
(2.14)
y
y
Therefore for any f : N → (0, ∞) with limn→∞ f (n) = ∞,
s
r Z ∞
2
p
f (n)
2
2
− x2
e
P0 sup Bt > nf (n) =
dx ∼
e− 2
1
π f 2 (n)
πf (n)
0≤t≤n
as n → ∞.
If we take f (n) = (2 + ) log n for any > 0, then
∞
X
sup Bt >
P0
p
nf (n) < ∞.
0≤t≤n
n=1
By Borel-Cantelli, a.s. sup0≤t≤n Bs ≤
is arbitrary, this implies
p
(2 + )n log n for all n sufficiently large. Since > 0
Bt
lim sup √
≤ 1.
2t log t
t→∞
To replace log t by log log t, we apply Borel-Cantelli along an exponentially increasing sequence
of times tn = αn , instead of tn = n. With f (t) = 2α log log t for any α > 1, we have
∞
X
n=1
P0
sup Bt >
p
αn f (αn ) < ∞.
0≤t≤αn
p
Therefore by Borel-Cantelli, a.s. sup0≤t≤αn Bt ≤ 2αn+1 log log αn for all n large. For any
t > 0, we can find n such that αn < t ≤ αn+1 . Then a.s. for all t sufficiently large,
s
p
sup0≤s≤αn+1 Bs
Bt
2αn+2 log log αn+1
log log αn+1
√
√
≤p
≤α
≤ α2 .
log log αn
2t log log t
2t log log t
2αn+2 log log αn+1
Since α > 1 can be chosen arbitrarily, this implies that
lim sup √
t→∞
Bt
≤1
2t log log t
a.s.
(2.15)
To prove the complimentary lower bound, we will find independent events and apply
Borel-Cantelli. As before, let tn = αn , except α > 1 will now be large. Note that
βf (t
P0 (Btn+1 −Btn
)
n+1
p
p
e− 2
> tn+1 f (tn+1 )) = P0 (Btn+1 −tn > tn+1 f (tn+1 )) ∼ √
2πβf (tn+1 )
tn+1
provided that βf (tn+1 ) ↑ ∞, where β = tn+1
−tn =
−2
f (tn ) ∼ 2β log n, and since β > 1, we have
∞
X
P0 (Btn+1 − Btn >
α
α−1 .
Take f (t) = 2β −2 log log t. Then
p
2β −2 tn+1 log log tn+1 ) = ∞.
n=1
6
as n → ∞,
Since (Btn+1 − Btn )n∈N are independent, by Borel-Cantelli, a.s. the event Btn+1 − Btn >
p
2β −2 tn+1 log log tn+1 occurs infinitely often. For such n sufficiently large, by (2.15) applied
to (−Bt )t≥0 , we have
p
Btn+1 > Btn + 2β −2 tn+1 log log tn+1
p
p
>
2β −2 tn+1 log log tn+1 − (2 + α−1 )tn log log tn .
Since β ↓ 1 as α ↑ ∞, and α > 1 can be made arbitrarily large, this implies that
lim sup √
t→∞
Bt
Btn
≥1
≥ lim sup √
2t log log t
2t
n→∞
n log log tn
a.s.,
which concludes the proof of (2.4).
References
[K97] O. Kallenberg. Foundations of Modern Probability. Springer, 1997.
[G72] A.M. Garsia. Continuity properties of Gaussian processes with multidimensional time
parameter. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and
Probability, Vol. II: Probability theory, 369–374. Univ. California Press, Berkeley, Calif.,
1972.
[GRR70] A.M. Garsia, E. Rodemich, and H. Jr. Rumsey. A real variable lemma and the
continuity of paths of some Gaussian processes. Indiana Univ. Math. Journal 20, 565-578,
1970/1971.
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