Convergence to equilibrium of Markov processes (eventually piecewise deterministic)

Transcription

1 Convergence to equilibrium of Markov processes (eventually piecewise deterministic) A. Guillin Université Blaise Pascal and IUF Rennes joint works with D. Bakry, F. Barthe, F. Bolley, P. Cattiaux, R. Douc, G. Fort, I. Gentil, F. Malrieu, P-A. Zitt A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

2 Introduction Our main goal: study the speed of convergence to equilibrium of Markov Process. Generator L Semigroup P t ergodic Invariant measure µ Carré du champ Γ(f, g) = 1 2 (L(fg) flg glf ). Energy E(f ) = flfdµ = Γ(f )dµ A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

3 Introduction Examples : Continuous time Markov chain: L = P I where P is the transition matrix. Reversible diffusion: L = V., Γ(f ) = f 2 Kinetic Fokker Planck: Lf (x, v) = v x f + v V (x). v f v. v f, Γ(f ) = v f 2 Hypocoercive walk: Lf (x, y) = y x f (x, y) + a(f (x, y) f (x, y)) A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

4 Introduction Examples (continued) : TCP process: with 0 < r < 1, l(x) = c or x Lf (x) = f (x) + l(x)(f (rx) f (x)) Ergodic Telegraph process: Lf (y, w) = w y f (y, w) + (a + (b a)1 {yw>0} )(f (y, w f (y, w)) Random switching: Lf (x, i) = L i f (x, i) + j a(i, j)(f (x, j) f (x, i)) What is the speed of convergence of P t to µ? in which distance? A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

5 Introduction Various techniques: Coupling TV distance, weighted TV distance, Wasserstein distance Lyapunov s conditions, control hitting times. Ad-Hoc Coupling Functional inequalities L 2 distance, entropy, Wasserstein distance Poincaré, log-sobolev inequalities, WJ, curvature... A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

6 Meyn-Tweedie s approach : Lyapunov condition and coupling Lyapunov condition: find W 1, a nice set C, b > 0 and a positive ϕ such that LW ϕ W + b 1 C. This condition expresses the strength which pushes the process to a nice region of the space. Remark: one may consider this condition as the supermartingale statement [ t ] [ t ] E x (W (X t )) + E x ϕ(x s )W (X s )ds W (x) + be x 1 C (X s )ds 0 0 A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

7 Examples Ornstein-Uhlenbeck : L = x.. W (x) = 1 + x 2, LW = 2n 2 x 2 W (x) + 2(n 1)1 { x 2 2n} but with another choice W (x) = e a x 2, LW = ( ( 2an + 4a a 1 ) ) x 2 W (x) 2 λ x 2 W (x) + b1 { x R} A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

8 Examples Ornstein-Uhlenbeck : L = x.. W (x) = 1 + x 2, LW = 2n 2 x 2 W (x) + 2(n 1)1 { x 2 2n} but with another choice W (x) = e a x 2, LW = ( ( 2an + 4a a 1 ) ) x 2 W (x) 2 λ x 2 W (x) + b1 { x R} A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

9 Examples Ornstein-Uhlenbeck : L = x.. W (x) = 1 + x 2, LW = 2n 2 x 2 W (x) + 2(n 1)1 { x 2 2n} but with another choice W (x) = e a x 2, LW = ( ( 2an + 4a a 1 ) ) x 2 W (x) 2 λ x 2 W (x) + b1 { x R} A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

10 Examples Exponential type process: L = x x.. choose a < 1 W (w) = e a x, LW c W (x) + b1 { x R} Cauchy type process: L = (n + α) V V. et V convexe. choose 2 < k < α(1 ε) + nε + 2 and small enough ε W (x) = 1 + x k, LW c (W (x)) k 2 k + b1 { x R} A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

11 Coupling by small set It is a probabilist approach: Lyapunov condition expresses that the process is pushed to nice regions how to express this strength? control of hitting times! Nice regions? small set A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

12 Coupling by small set It is a probabilist approach: Lyapunov condition expresses that the process is pushed to nice regions how to express this strength? control of hitting times! Nice regions? small set A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

13 Coupling by small set It is a probabilist approach: Lyapunov condition expresses that the process is pushed to nice regions how to express this strength? control of hitting times! Nice regions? small set A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

14 Mathematically: We have to build two processes X t and Y t such that: X t and Y t has the same law but starting from x and y after some random time T, the two processes sticks together Indeed, P t (x, ) P t (y, ) TV inf couplage E(1 X t Y t ) P(T > t) For the coupling construction we will use a minorization condition and to control T the Lyapunov condition. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

15 Coupling construction Minorization condition: Suppose that for some set C and t > 0, there exists ε > 0 such thate (Minor) x C, P t (x, ) εν( ) It enables to write a mixture of probability for P t : if x C P t (x, ) = ε ν( ) + (1 ε) P t (x, ) ε ν( ) 1 ε Intuition : if my process is in C there is a probability ε to be distributed at time t as ν independently of my position in C! A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

16 Definition of (X t, Y t ) 1 X 0 = x, Y 0 = y. 2 If t 0 = inf{t; (X t, Y t ) C C} and t n = inf{t t n 1 + t ; (X t, Y t ) C C}. If we have not coupled at t i then: with probability ε, X ti +t = Y t i +t = Z with Z ν and we have coupled, then set T = t i + t!. with probability 1 ε, generate X ti +t et Y t i +t independently with the residual kernel We need to control the return times in C C. Remark that before coupling, the two processes are generated by L L. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

17 Denote τ C (t ) = inf{t t ; X t C}. Suppose the Lyapunov conditions LW δ W + b1 C then x C, E x (e δτ C (0) ) W (x) or the weaker: with strictly concave ϕ, b > 0 such that LW ϕ(w ) + b1 C with H ϕ (u) = u 1 ( x C, E x H 1 ϕ (τ C (t ) ) W (x) + c b,ϕ,θ 1 ϕ ds. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

18 Denote τ C (t ) = inf{t t ; X t C}. Suppose the Lyapunov conditions LW δ W + b1 C then x C, E x (e δτ C (0) ) W (x) or the weaker: with strictly concave ϕ, b > 0 such that LW ϕ(w ) + b1 C with H ϕ (u) = u 1 ( x C, E x H 1 ϕ (τ C (t ) ) W (x) + c b,ϕ,θ 1 ϕ ds. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

19 Denote τ C (t ) = inf{t t ; X t C}. Suppose the Lyapunov conditions LW δ W + b1 C then x C, E x (e δτ C (0) ) W (x) or the weaker: with strictly concave ϕ, b > 0 such that LW ϕ(w ) + b1 C with H ϕ (u) = u 1 ( x C, E x H 1 ϕ (τ C (t ) ) W (x) + c b,ϕ,θ 1 ϕ ds. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

20 Denote τ C (t ) = inf{t t ; X t C}. Suppose the Lyapunov conditions LW δ W + b1 C then x C, E x (e δτ C (0) ) W (x) or the weaker: with strictly concave ϕ, b > 0 such that LW ϕ(w ) + b1 C with H ϕ (u) = u 1 ( x C, E x H 1 ϕ (τ C (t ) ) W (x) + c b,ϕ,θ 1 ϕ ds. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

21 Proof Let apply Itô s formula e t W (x): ( ) ( E x e at τ C Ex e at τ C W (X t τc ) ) ) ( t τc ) W (x) + E x (aw + LW )(X s )e as ds 0 ( t τc ) W (x) + E x (a δ)w (X s )e as ds 0 W (x) if a δ. The subexponential case is a little bit more complicated. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

22 With the Lyapunov condition LW δ W + b1 C and the minorization condition with C = {W K} then for ρ = ρ(ε, C, t, W ) < 1 K > 1, P t (x, ) P t (y, ) TV Kρ t (W (x) + W (y)) and if for ϕ strictly concave, b > 0such that then for K = K(ε, C, ϕ, W ) LW ϕ(w ) + b1 C P t (x, ) P t (y, ) TV K Hϕ 1 (W (x) + W (y)) (t t ) A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

23 Advantages generic method (non reversible, discrete, discrete/continuous) sharp qualitative results Disadvantages poor quantitative results because of the minorization condition can be difficult to find the Lyapunov function W (oscillators,...). A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

24 We thus often have to consider ad-hoc coupling deduced from the dynamics or other distances: for example W p p (ν, µ) := inf {E(d(X, Y ) p ; X ν, Y µ} 2-Wasserstein distance and synchronous coupling (convexity) 1-Wasserstein distance and reflection coupling (convexity at infinity) Advantages: work for hypoelliptic diffusions, non linear process such as McKean-Vlasov models. For more examples see the talks of Bardet, Cloez, Guerin... A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

25 Functional inequalities Functional inequalities Let us recall some well known facts: Indeed L 2 exponential decay Poincaré inequality C Var µ (f ) Γ(f )dµ d dt Var µ(p t f ) = 2 P t flp t fdµ = 2E(P t f ) 2C Var µ (P t f ) and Gronwall s lemma gives Var µ (P t f ) e 2Ct Var µ (P t f ). Entropic decay log-sobolev inequality (Diffusion) C Ent µ (f 2 ) 2 Γ(f )dµ A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

26 Functional inequalities How to get a Poincaré or log-sobolev inequality? One well known criterion: Γ 2 criterion. BE(K, ) : Γ 2 (f ) K Γ(f ) where Γ 2 (f, g) := 1 2 (LΓ(f, g) γ(f, Lg) Γ(g, Lf )). Theorem : BE(K, ) = log-sobolev and Poincaré inequality with constant K. Example: L = V., Γ 2 (f ) = Hessf f t HessV f BE(K, ) HessV K Id. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

27 Functional inequalities Indeed: BE(K, ) Γ(P t f ) e 2Kt P t Γ(f ) (Trick show ψ (s) 0, with ψ(s) = P s (Γ(P t s f )). After that: Var µ (f ) = s (P s f ) 2 dsdµ 0 = 2 Γ(P s f )dµds 0 c 2 e 2Ks ds P s Γ(f )dµ = 1 K 0 Γ(f )dµ Commutation is crucial (in this proof...), but one can give integrated criterion. See the talk of Joulin for refinements around this commutation property. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

28 Functional inequalities Indeed: BE(K, ) Γ(P t f ) e 2Kt P t Γ(f ) (Trick show ψ (s) 0, with ψ(s) = P s (Γ(P t s f )). After that: Var µ (f ) = s (P s f ) 2 dsdµ 0 = 2 Γ(P s f )dµds 0 c 2 e 2Ks ds P s Γ(f )dµ = 1 K 0 Γ(f )dµ Commutation is crucial (in this proof...), but one can give integrated criterion. See the talk of Joulin for refinements around this commutation property. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

29 Functional inequalities Links between functional inequalities and Poincaré inequalities? In fact, in the reversible case µ(flg) = µ(glf ), one has in regular situation Lyapunov Condition Poincaré inequalities Indeed let us suppose that LW λw + b1 C and a local Poincaré inequality (f µ(f 1 C )) 2 dµ κ C Γ(f )dµ C then Poincaré inequality holds with constant 1 λ (1 + bκ C ) C A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

30 Functional inequalities Proof: Var µ (f ) (f µ(f 1 C )) 2 dµ 1 (f µ(f 1 C )) 2 LW λ W dµ + 1 λ C (f µ(f 1 C )) 2 dµ Use then local Poincaré inequality and the large deviations bound (or direct calculus for diffusion) in the reversible case f 2 LW W dµ Γ(f )dµ A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

31 Functional inequalities There are also variants for subexponential decay via weak Poincaré decay in entropy via logarithmic Sobolev inequality decay in Wasserstein distance via WJ inequality in L 2 weighted space in non reversible case via Lyapunov-Poincaré inequalities A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

32 Functional inequalities Advantages generic method with other consequences (concentration,...) variety of criterion Disadvantages non reversible case is difficult, a refined spectral analysis may be used (see Monmarché) sharp constants can be hard to get. A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

33 Conclusion Conclusion Two different methods which can be generically tested, at least for qualitative results... In general, one has to refine these methods to get a good answer to a particular problem... it will be the subject of quite a few talks here... A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26

34 Conclusion Facultative slide : let consider P t the Heat semigroup on R n. By synchronous coupling it is not difficult to get that for densities (wrt Lebesgue) f, g W 2 (P t f, P t g) W 2 (f, g) but we recently prove by analytical method (with Bolley and Gentil) that W 2 2 (P t f, P t g) W 2 2 (f, g) 2 n Can you find a coupling proof???? t 0 (Ent µ (P s f ) Ent µ (P s g)) 2 ds A. Guillin (UBP/IUF) Convergence to equilibrium 15/05/ / 26