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math4512 From kinetic equations to statistical mechanics
Preface
These lecture notes have been prepared for a series of lectures to be given at the Summer School
“From kinetic equations to statistical mechanics”, organised by the Henri Lebesgue Center in
Saint Jean de Monts, from June 28th to July 2nd 2021. This is a preliminary version of the notes,
which may still contain errors. A third chapter, concerning the theory of large deviations and
its applications to stochastic differential equations, has yet to be written.
The author wishes to thank the organisers of the Summer School, Frédéric Hérau (Univer-
sité de Nantes), Laurent Michel (Université de Bordeaux), and Karel Pravda-Starov (Université
de Rennes 1) for the kind invitation that provided the motivation to compile these lecture
notes.
iii
iv Contents
Chapter1
Stochastic Differential Equations and
Partial Differential Equations
1.1 Brownian motion
The fundamental building block of the theory of stochastic differential equations is a math-
ematical object called Wiener process, or Brownian motion. This should not be confused with
the physical phenomenon of Brownian motion, describing for instance the erratic movements
of a small particle in a fluid, though the mathematical model has of course been introduced
as a simplified description of the physical process. There is a huge literature on properties of
Brownian motion. In what follows, we will focus on only a few of these properties that will be
important for links between stochastic and partial differential equations.
1.1.1 Construction of Brownian motion
Heuristically, Brownian motion can be defined as a scaling limit of a random walk. Let {Xn}n>0
be a symmetric random walk on Z, defined as
Xn =
n∑
i=1
ξi ,
where the ξi are i.i.d. (independent and identically distributed) random variables, taking val-
ues ±1 with probability 12 . The following properties are easy to check:
1. Xn has zero expectation: E[Xn] = 0 for all n;
2. The variance of Xn satisfies Var(Xn) = n;
3. Xn takes values in {−n,−n+ 2, . . . ,n− 2,n}, with
P
{
Xn = k
}
=
1
2n
n!(n+k
2
)
!
(n−k
2
)
!
.
4. Independent increments: for all n > m> 0, Xn −Xm is independent of X1, . . . ,Xm;
5. Stationary increments: for all n > m> 0, Xn −Xm has the same distribution as Xn−m.
Consider now the sequence of processes
W
(n)
t =
1√
n
Xbntc , t ∈ R+ , n ∈ N .
At stage n, space has been compressed by a factor n, while time has been sped up by a factor√
n (Figure 1.1).
1
2 Chapter 1. Stochastic Differential Equations and Partial Differential Equations
5 10
−20−2
−1
0
1
2
n
Sn
10 20 30 40 50−20
−4
−2
0
2
4
n
Sn
50 100 150 200 250
−20
−10
0
10
n
Sn
500 1000 1250
−60
−40
−20
0
20
n
Sn
Figure 1.1 – Two realisations (one in red, the other one in blue) of a symmetric random walk on Z,
seen at different scales. From one picture to the next, the horizontal scale is compressed by a factor
5, while the vertical scale is compressed by a factor
√
5.
Formally, as n→∞, the processes {W (n)t }t>0 should converge to a stochastic process {Wt}t>0
satisfying the following properties.
1. E[Wt] = 0 for all t> 0;
2. The variance of Wt satisfies
Var(Wt) = limn→∞
(
1√
n
)2
bntc = t .
3. By the central limit theorem, Xbntc/
√bntc converges in distribution to a standard normal
random variable. Therefore, for each t, Wt follows a normal law N (0, t).
4. Independent increments: for all t > s> 0, Wt −Ws est independent of {Wu}06u6s;
5. Stationary increments: for all t > s> 0, Wt −Ws has the same distribution as Wt−s.
This motivates the following definition.
Definition 1.1.1: Brownian motion
Standard Brownian motion (also called the standard Wiener process) is the stochastic process
{Wt}t>0 satisfying:
1. W0 = 0;
2. Independent increments: for all t > s> 0, Wt −Ws est independent of {Wu}u6s;
3. Stationary increments: for all t > s> 0, Wt −Ws follows a normal law N (0, t − s).
Theorem 1.1.2: Existence of Brownian motion
There exists a stochastic process {Wt}t>0 satisfying Definition 1.1.1, and whose trajectories
t 7→ Bt(ω) are continuous.
Proof:
1.1. Brownian motion 3
W
(3)
t
W
(2)
t
W
(1)
t
W
(0)
t
X1/2
X1
X3/4
X1/4
1
4
1
2
3
4
1
Figure 1.2 – Construction of Brownian motion by interpolation.
1. We start by constructing {Wt}06t61 from a collection of independent Gaussian random vari-
ables V1,V1/2,V1/4,V3/4,V1/8, . . . , all with zero mean, where V1 and V1/2 have variance 1 and
each Vk2−n has variance 2−(n−1) (k < 2n odd).
We first show that if Xs et Xt are two random variables such that Xt−Xs is centred, Gaussian
with variance t − s, then there exists a random variable X(t+s)/2 such that the random vari-
ables Xt −X(t+s)/2 and X(t+s)/2 −Xs are i.i.d. with law N (0, (t − s)/2). If U = Xt −Xs and V is
independent of U , with the same distribution, il suffices to define X(t+s)/2 by
Xt −X(t+s)/2 = U +V2
X(t+s)/2 −Xs = U −V2 . (1.1.1)
Indeed, it is easy to check that these variables have the required distributions, and that they
are independent, since E[(U +V )(U −V )] = E[U2]−E[V 2] = 0, and normal random variables
are independent if and only if they are uncorrelated.
Let us set X0 = 0, X1 = V1, and construct X1/2 with the above procedure, taking V = V1/2.
Then we construct X1/4 with the help of X0, X1/2 and V1/4, and so on, to obtain a family of
variables {Xt}t=k2−n,n>1,k<2n such that for t > s, Xt −Xs is independent of Xs and has distribu-
tion N (0, t − s).
2. For n> 0, let {W (n)t }06t61 be the stochastic process with piecewise linear trajectories on in-
tervals [k2−n, (k+1)2−n], k < 2n, and such thatW (n)k2−n = Xk2−n (Figure 1.2). We want to show
that the sequence W (n)(ω) converges uniformly on [0,1] for any realisation ω of the Vi . We
thus have to estimate
∆(n)(ω) = sup
06t61
∣∣∣W (n+1)t (ω)−W (n)t (ω)∣∣∣
= max
06k62n−1
max
k2−n6t6(k+1)2−n
∣∣∣W (n+1)t (ω)−W (n)t (ω)∣∣∣
= max
06k62n−1
∣∣∣∣X(2k+1)2−(n+1)(ω)− 12(Xk2−n(ω) +X(k+1)2−n(ω))∣∣∣∣
(see Figure 1.3). The term in the absolute value is 12V(2k+1)2−(n+1) by construction, c.f. (1.1.1),
4 Chapter 1. Stochastic Differential Equations and Partial Differential Equations
W
(n+1)
t
W
(n)
t
X(2k+1)2−(n+1)
X(k+1)2−n
Xk2−n+X(k+1)2−n
2
Xk2−n
k2−n (2k + 1)2−(n+1) (k + 1)2−n
Figure 1.3 – Computation of ∆(n).
which is Gaussian with variance 2−n. Therefore,
P
{
∆(n) >
√
n2−n} = P{ max
06k62n−1
∣∣∣V(2k+1)2−(n+1) ∣∣∣> 2√n2−n}
6 2 · 2n
∫ ∞
2
√
n2−n
e−x2/2·2−n dx√
2pi2−n
= 2 · 2n
∫ ∞
2
√
n
e−y2/2 dx√
2pi
6 const2n e−2n ,
and thus ∑
n>0
P
{
∆(n) >
√
n2−n}6 const ∑
n>0
(2e−2)n <∞ .
The Borel–Cantelli lemma shows that with probability 1, there exist only finitely many n
for which ∆(n) >
√
n2−n. It follows that
P
{∑
n>0
∆(n) <∞
}
= 1 .
The sequence {W (n)t }06t61 is thus a Cauchy sequence for the sup norm with probability 1,
and therefore converges uniformly. For t ∈ [0,1] we set
W 0t =
limn→∞W (n)t if the sequence converges uniformly0 otherwise (with probability 0).
It is easy to check that B0 satisfies the three properties of the definition.
3. To extend the process to all times, we build independent copies {W i}i>0 and set
Wt =
W 0t 06 t < 1
W 01 +W
1
t−1 16 t < 2
W 01 +W
1
1 +W
2
t−2 26 t < 3
. . .
This concludes the proof.
1.1. Brownian motion 5
Remark 1.1.3: n-dimensional Brownian motion
For any n ∈ N, one can define n-dimensional Brownian motion in the same way as in Defini-
tion 1.1.1, except that the normal laws are n-dimensional. Its components are then simply
independent 1-dimensional Brownian motions.
1.1.2 Basic properties of Brownian motion
The following basic properties of Brownian motion follow more or less immediately from Def-
inition 1.1.1.
1. Markov property: For any Borel set A ⊂ R,
P
{
Wt+s ∈ A
∣∣∣∣Wt = x} = ∫
A
p(t + s,y|t,x)dy ,
independently of {Wu}u
−(y−x)2/2s
√
2pis
. (1.1.2)
The proof follows directly from the decompositionWt+s =Wt+(Wt+s−Wt), where the second
term is independent of the first one, with distribution N (0, s). In particular, one checks the
Chapman–Kolmogorov equation: For t > u > s,
p(t,y|s,x) =
∫
R
p(t,y|u,z)p(u,z|s,x)du . (1.1.3)
2. Differential property: For all t>0, {Wt+s−Wt}s>0 is a standard Brownian motion, independent
of {Wu}u
4. Symmetry: {−Wt}t>0 is a standard Brownian motion.
5. Gaussian process : The Wiener process is Gaussian with zero mean (meaning that its finite-
dimensional joint distributions P{Wt1 6 x1, . . . ,Wtn 6 xn} are centred normal), and charac-
terised by its covariance
Cov{Wt ,Ws} ≡ E[WtWs] = s∧ t (1.1.4)
(where s∧ t denotes the minimum of s and t).
Proof: For s < t, we have
E[WtWs] = E[Ws(Ws +Wt −Ws)] = E[W 2s ] +E[Ws(Wt −Ws)] = s ,
since the second term vanishes by the independent increments property.
In fact, one can show that a centred Gaussian process whose covariance satisfies (1.1.4) is a
standard Wiener process.
One important consequence of the scaling and independent increments properties is then
the following.
Theorem 1.1.4: Non-differentiability of Brownian paths
The paths t 7→Wt(ω) are almost surely nowhere Lipschitzian, and thus nowhere differen-
tiable.
6 Chapter 1. Stochastic Differential Equations and Partial Differential Equations
Proof: Fix C <∞ and introduce, for n> 1, the event
An =
{
ω : ∃s ∈ [0,1] s.t. |Wt(ω)−Ws(ω)|6C|t − s| if |t − s|6 3n
}
.
We have to show that P(An) = 0 for all n. Observe that if n increases, the condition gets weaker,
so that An ⊂ An+1. For n> 3 and 16 k6n− 2, define
Yk,n(ω) = max
j=0,1,2
{∣∣∣∣W(k+j)/n(ω)−W(k+j−1)/n(ω)∣∣∣∣} ,
Bn =
n−2⋃
k=1
{
ω : Yk,n(ω)6
5C
n
}
.
The triangular inequality implies An ⊂ Bn. Indeed, let ω ∈ An. If for instance s = 1, then for
k = n− 2, one has∣∣∣W(n−3)/n(ω)−W(n−2)/n(ω)∣∣∣6 ∣∣∣W(n−3)/n(ω)−W1(ω)∣∣∣+ ∣∣∣W1(ω)−W(n−2)/n(ω)∣∣∣6C(3n + 2n)
and thus ω ∈ Bn. It follows from the independent increments and scaling properties that
P(An)6P(Wn)6nP
(
|B1/n|6 5Cn
)3
= nP
(
|W1|6 5C√
n
)3
6n
(
10C√
2pin
)3
.
Therefore P(An)→ 0 for all n→∞. But since P(An)6 P(An+1) for all n, this implies P(An) = 0
for all n.
Remark 1.1.5: Hölder regularity of Brownian paths
Even though paths of Brownian motion are nowhere differentiable, one can show that they
do have a regularity that is better than continuity: namely, the paths are almost surely
(locally) Hölder continuous of exponent α for any α < 12 . This can be shown by applying
the Kolmogorov–Centsov continuity criterion.
1.1.3 Brownian motion and heat equation
Observe that the Gaussian transition probabilities (1.1.3) of the Wiener process are, up to a
scaling, equal to the heat kernel. In particular, p(t,x|0,0) satisfies the heat equation
∂
∂t
p(t,x|0,0) = 1
2
∆p(t,x|0,0)
p(0,x|0,0) = δ(x) ,
where we write ∆ for the second derivative with respect to x. This reflects the fact that paths
of Brownian motion have the same diffusive behaviour as solutions of the heat equation.
Similarly, transition probabilities of n-dimensional Brownian motion are given by
p(t + s,y|t,x) = e
−‖y−x‖2/2s
(2pis)n/2
,
and satisfy therefore the n-dimensional heat equation.
It is, however, important to realise that Brownian motion contains much more information
than the solutions of the heat equation, since it gives a probability distribution on paths t 7→
Wt(ω), rather than just a collection of probability distributions for the Wt(ω) with t > 0. To
illustrate the difference, we discuss two examples of modifications of Brownian motion.
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