Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Controllability implies ergodicity
Armen Shirikyan
Department of Mathematics
University of Cergy–Pontoise
Infinite-dimensional dynamics, dissipative systems, and attractors
Lobachevsky State University of Nizhny Novgorod
13 – 17 July 2015
1 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Two examples
Example
Let us consider the following ODE on a compact Riemannian
manifold X :
u̇ = V0 (u) +
m
X
η j (t)Vj (u),
u(t) ∈ X .
(1)
j=1
Here V0 , . . . , Vm are smooth vector fields on X and η j (t) are
scalar random processes. Under mild regularity assumptions
on η j , the Cauchy problem for (1) is well posed, and the
solutions are random processes with range in X .
2 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Two examples
Example
Let us consider the Navier–Stokes system in a bounded
domain D ⊂ R2 :
∂t u + hu, ∇iu − ν∆u + ∇p = h(x),
u ∂D = η.
div u = 0,
x ∈ D,
(2)
Here η is an R2 -valued sufficiently regular random field
on R+ × ∂D. Assuming that η is bounded, on can prove that the
random flow generated by (2) possesses a compact invariant
absorbing set X ⊂ L2 (D, R2 ). For any initial state u0 ∈ X , the
corresponding solution of (2) is a random process in X .
Problem: Asymptotic behaviour of solutions as t → ∞.
3 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Outline
Total variation distance
Mixing in total variation
General criterion
Application to ODE’s with random coefficients
Kantorovich–Wasserstein metric
Weak mixing
General criterion
Application to the Navier–Stokes system
Open problems
4 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Total variation distance
Let X be a compact metric space, P(X ) the space of probability
measures on X , and D(ξ) the law of a random variable ξ.
Definition
For any µ1 , µ2 ∈ P(X ), define the total variation distance
kµ1 − µ2 kvar = sup |µ1 (Γ) − µ2 (Γ)|,
Γ∈B(X )
where B(X ) is the set of Borel subsets of X .
Lemma
For any µ1 , µ2 ∈ P(X ), there are µ, µ01 , µ02 ∈ P(X ) such that
µ1 = (1 − d)µ + dµ01 ,
µ2 = (1 − d)µ + dµ02
(3)
where d = kµ1 − µ2 kvar .
5 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Maximal coupling
Corollary
For any µ1 , µ2 ∈ P(X ), there is a pair of X -valued random
variables (ξ1 , ξ2 ) (called maximal coupling) such that
D(ξ1 ) = µ1 ,
D(ξ2 ) = µ2 ,
(4)
P{ξ1 6= ξ2 } = kµ1 − µ2 kvar .
(5)
Moreover, for any pair (ξ1 , ξ2 ) satisfying (4) we have
P{ξ1 6= ξ2 } ≥ kµ1 − µ2 kvar .
(6)
Remark
The corollary shows that the total variation distance between
two measures µ1 and µ2 is the minimal probability with which
two random variables with those laws must be different.
6 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Discrete-time random dynamical systems
Let X be a compact metric space, E separable Hilbert space,
and S : X × E → X a continuous mapping. Consider the RDS
uk = S(uk −1 , ηk ),
k ≥ 1,
(7)
where {ηk } is a sequence of i.i.d. random variables in E.
Example
Consider an ODE on a compact Riemannian manifold X :
u̇ = V0 (u) +
m
X
η j (t)Vj (u).
(8)
j=1
Setting η = (η 1 , . . . , η m ) and ηk = η|[k −1,k ] , we write
u(k ) = S u(k − 1), ηk , S : u0 , η [0,1] 7→ u(1).
7 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Main result
Define the transition function
Pk (v , Γ) = Pv {uk ∈ Γ},
v ∈ X,
Γ ∈ B(X ).
Theorem
Suppose ∃ û ∈ X ∃ r > 0 with the following properties:
Recurrence: ∃ m ≥ 1 ∃ p > 0 such that
Pm u, BX (û, r ) ≥ p ∀u ∈ X .
(9)
Coupling: ∃ ε > 0 such that
kP1 (u, ·) − P1 (u 0 , ·)kvar ≤ 1 − ε
∀u, u 0 ∈ BX (û, r ).
(10)
Then ∃ C, γ > 0 and a unique measure µ ∈ P(X ) such that
kDλ (uk ) − µkvar ≤ C e−γk
∀k ≥ 0
∀λ ∈ P(X ).
(11)
8 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Differential equations with random coefficients
Consider an ODE on a compact Riemannian manifold X :
u̇ = V0 (u) +
m
X
η j (t)Vj (u).
(12)
j=1
We assume that
1
m
η(t) = η (t), . . . , η (t) =
∞
X
I[k −1,k ) (t)ηk (t − k + 1),
(13)
k =1
where {ηk } are i.i.d. random variables in E := L2 (0, 1; Rm ).
The Cauchy problem for (12) is well posed, and all solutions
exist globally in time for any initial state v ∈ X .
9 / 20
Introduction
Total variation
Theorem
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Main result for ODE’s
Assume that the following two hypotheses are satisfied:
Hörmander condition: Lieu (V1 , . . . , Vm ) = Tu X for any u ∈ X .
Non-degeneracy: There is an ONB {ei } in E such that
ηk (t) =
∞
X
ξik ei (t)
(14)
i=1
where {ξik } are independent random variables satisfying
D(ξik ) = ρi (r )dr ;
ρi ∈ C(R),
ρi > 0;
∞
X
E ξik2 < ∞.
i=1
Then ∃ a unique µ ∈ P(X ) and ∃ C, γ > 0 such that
kDλ (u(k )) − µkvar ≤ C e−γk
∀k ≥ 0 ∀λ ∈ P(X ).
(15)
10 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Kantorovich–Wasserstein and total variation metrics
Define the space of uniformly continuous bounded functions
Lb (X ) = {f ∈ Cb (X ) : |f (u) − f (v )| ≤ C dX (u, v ) ∀ u, v ∈ X }.
Denote by k · kL the natural norm on Lb (X ) and define
Z
kµ1 − µ2 k∗L = sup |(f , µ1 ) − (f , µ2 )|, (f , µ) =
f dµ.
kf kL ≤1
X
It can be proved that
kµ1 − µ2 kvar =
1
sup |(f , µ1 ) − (f , µ2 )|,
2 kf k∞ ≤1
so that the KW metric is (much) weaker than the TV metric.
Example
If u, v ∈ X and u 6= v , then
kδu − δv kvar = 1,
kδu − δv k∗L = dX (u, v ).
11 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
General criterion
Consider the random dynamical system
uk = S(uk −1 , ηk ),
uk ∈ X ,
ηk ∈ E.
Theorem
Suppose the following two conditions hold for some û ∈ X :
Recurrence: For any δ > 0 there are p > 0, m ≥ 1 such that
Pm (u, B(û, δ)) ≥ p
for any u ∈ X .
(16)
Stability: ∃ function ε(r ) > 0 going to zero as r → 0+ such that
sup kPk (u, ·) − Pk (u 0 , ·)k∗L ≤ ε(r ) ∀ u, u 0 ∈ BX (û, r ).
(17)
k ≥0
Then there is a unique µ ∈ P(X ) such that
sup kDλ (uk ) − µk∗L → 0 as k → ∞.
(18)
λ∈P(X )
12 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Sufficient condition for stability
Proposition
Suppose the following condition holds for some C and q < 1:
Squeezing: For any u, u 0 ∈ X there are X -valued random
variables v , v 0 such that
P dX (v , v 0 ) ≤ q dX (u, u 0 ) ≥ 1 − C dX (u, u 0 ),
D(v ) = P1 (u, ·),
0
0
D(v ) = P1 (u , ·).
(19)
(20)
Then the stability holds for any û ∈ X .
Remark
It can be proved that if the recurrence and squeezing conditions
are satisfied, then we have the exponential convergence
sup kDλ (uk ) − µk∗L ≤ C e−γk
∀k ≥ 0.
λ∈P(X )
13 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Idea of the proof of the proposition
Let us denote by R(u, u 0 ) and R0 (u, u 0 ) the random variables v
and v 0 satisfying (19) and (20). Define a sequence (vk , vk0 ) by
v0 = u,
v00 = u 0 ,
vk = R(vk −1 , vk0 −1 ),
vk0 = R0 (vk −1 , vk0 −1 ).
Then D(vk ) = Pk (u, ·), D(vk0 ) = Pk (u 0 , ·) for k ≥ 0. Moreover,
P{dX (vk , vk0 ) ≤ 2−k dX (u, u) for all k ≥ 0} ≥ 1−C ku−u 0 k. (21)
For any bounded 1-Lipschitz function f : X → R, we have
f , Pk (u, ·) − f , Pk (u 0 , ·) = E f (vk ) − f (v 0 ) k
= E f (vk ) − f (v 0 ) (IG + IGc )
k
−k
≤2
0
dX (u, u ) + C dX (u, u 0 ).
14 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Navier–Stokes system with a boundary noise
Let D ⊂ R2 and D1 = (0, 1) × D. Consider the problem
∂t u + hu, ∇iu − ν∆u + ∇p = h(t, x),
u ∂D = η(t, x),
div u = 0,
u(0, x) = u0 (x).
(22)
(23)
(24)
1 (R × D) and
Here h is a 1-periodic function belonging to Hloc
+
η(t, x) is a space-time localised noise of the form
η(t, x) =
∞
X
I[k −1,k ) (t)η(t − k + 1, x),
(25)
k =1
where {ηk } is a sequence i.i.d. random variables in the space
Z
n
o
5/2
E = g∈H
(0, 1) × ∂D :
hg(t), nidσx ≡ 0, supp g ⊂ Q ,
∂D
where Q b (0, 1) × ∂D.
15 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Navier–Stokes system with a boundary noise
The restrictions of trajectories of (22), (23) to Z+ form a random
dynamical system in the space
H = {u ∈ L2 (D, R2 ) : div u = 0 in D, hu, ni = 0 on ∂D}.
Structure of the noise. ∃ ONB {ej } in the space E such that
ηk =
∞
X
bj ξjk ej (t, x)
(26)
j=1
where {bj } ⊂ R is a sequence going to zero sufficiently fast
and {ξjk } are independent random variables such that
D(ξjk ) = ρj (r ) dr ,
ρj ∈ C 1 (R),
supp ρj ⊂ [−1, 1].
The support K of the law of ηk is a compact subset in E.
16 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Mixing for the Navier–Stokes system
Theorem
Assume that the following condition holds for some û ∈ H:
Approximate controllability to û: For any R, δ > 0 ∃ m ≥ 1 such
that, given u0 ∈ BH (R) one can find η1 , . . . ηm ∈ K for which the
corresponding solution of (22), (23) satisfies the inequality
ku(k ) − ûk ≤ δ.
In this case, there is N ≥ 1 depending on ν > 0 such that if
bj 6= 0 for 1 ≤ j ≤ N,
(27)
then ∃ C, γ > 0 and a unique measure µ ∈ P(H) such that
Z
∗
−γk
kDλ (u(m)) − µkL ≤ C e
1+
kukλ(du)
∀λ ∈ P(H).
H
17 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Open problems
Approximate controllability by a bounded localised force
Our result on mixing of the flow for the Navier–Stokes system
requires global approximate controllability by a control taking
values in the support of η. This condition is trivially satisfied if
h ≡ 0, and the support of η contains zero. Due to
Coron–Fursikov–Imanuvilov (1996–1999), the problem is
globally approximately controllable by an unbounded C ∞ force.
The following question remains completely open:
Nontrivial uncontrolled dynamics
Given a smooth function h, find a compact (or even bounded)
subset K ⊂ E and a point û ∈ H such that the Navier–Stokes
system is globally approximately controllable to û with a
K-valued control.
18 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
Open problems
Squeezing by a bounded Fourier-localised force
Let us consider the Navier–Stokes system with the RHS
f (t, x) = h(t, x) +
N
X
η j (t)ej (x),
(28)
j=1
where ej is an ONB in H. Due to Agrachev–Sarychev (2005),
the problem is globally approximately controllable for any ν > 0.
The following question is very important when dealing with the
problem of mixing for a Fourier-localised noise:
Local stabilisation
Given an arbitrary right-hand side h(t, x) and initial points
v , v 0 ∈ H, construct functions η 1 , . . . , η N ∈ L2 (0, 1) such that
ku(1) − u 0 (1)k ≤ qkv − v 0 k,
kη j kL2 ≤ Ckv − v 0 kα ,
(29)
where q < 1, α ≤ 1, and C are some positive numbers.
19 / 20
Introduction
Total variation
Mixing in total variation
Kantorovich–Wasserstein metric
Weak mixing
Open problems
References
• General existence theory: Hopf (1952), Foiaş (1972–73),
Vishik–Fursikov–Komech (1976–80);
• Rough noise: Flandoli–Maslowski (1995),
Bricmont–Kupiainen–Lefevere (2001),
Goldys–Maslowski (2005), many other works.
• Noise effective in determining modes:
Kuksin–AS (2000), E–Mattingly–Sinai (2001),
Bricmont–Kupiainen–Lefevere (2002), many other works.
• Highly degenerate noise:
Hairer–Mattingly (2006–2011): h ≡ 0, D = T2 ;
AS (2015): spatially localised noise with arbitrary D ⊂ R2 ;
Földes–Glatt-Holtz–Richards–Thomann (2015):
Boussinesq system with h ≡ 0, D = T2 and noise acting
only on the equation for temperature.
• Kuksin–AS: Mathematics of 2D Turbulence (2012).
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