Weak Feller Property and Invariant Measures

Transcription

1 Weak Feller Property and Invariant Measures Martin Ondreját, J. Seidler, Z. Brzeźniak Institute of Information Theory and Automation Academy of Sciences Prague September 11, 2012

2 Outline 2010: Stochastic wave equations polynomial nonlinearities Riemannian manifolds existence uniqueness temporal regularity 2012: Second order (in time) equations sequential weak Feller property invariant measures wave and beam equation

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4 Markov semigroup, Invariant measure SPDE in a separable Hilbert space X du = (Au + f (u)) dt + g(u) dw weak existence, uniqueness in law p(t, x, J) = P x [u x (t) J], J B(X) P t h(x) = h dp t,x, Pt ν(j) = p(t, x, J) dν X X (P t ) is a semigroup on B(X), (Pt ) is a semigroup on P(X) µ is invariant iff Pt µ = µ for every t 0

5 Markov semigroup, Invariant measure SPDE in a separable Hilbert space X du = (Au + f (u)) dt + g(u) dw weak existence, uniqueness in law p(t, x, J) = P x [u x (t) J], J B(X) P t h(x) = h dp t,x, Pt ν(j) = p(t, x, J) dν X X (P t ) is a semigroup on B(X), (Pt ) is a semigroup on P(X) µ is invariant iff Pt µ = µ for every t 0

6 Markov semigroup, Invariant measure SPDE in a separable Hilbert space X du = (Au + f (u)) dt + g(u) dw weak existence, uniqueness in law p(t, x, J) = P x [u x (t) J], J B(X) P t h(x) = h dp t,x, Pt ν(j) = p(t, x, J) dν X X (P t ) is a semigroup on B(X), (Pt ) is a semigroup on P(X) µ is invariant iff Pt µ = µ for every t 0

7 Markov semigroup, Invariant measure SPDE in a separable Hilbert space X du = (Au + f (u)) dt + g(u) dw weak existence, uniqueness in law p(t, x, J) = P x [u x (t) J], J B(X) P t h(x) = h dp t,x, Pt ν(j) = p(t, x, J) dν X X (P t ) is a semigroup on B(X), (Pt ) is a semigroup on P(X) µ is invariant iff Pt µ = µ for every t 0

8 Krylov Bogolyubov Tightness ν s.t. {P t ν : t 0} tight + Feller P t h C b (X) for h C b (X) = existence of an invariant measure Barbu, Da Prato 02 + Tubaro 06 : Wave + additive noise Ichikawa 84 : Take (X, w) instead of (X, ) Maslowski, Seidler 99,01 : Take (X, bw) instead of (X, w) Brzeźniak, Ondreját, Seidler : Wave, beam equation

9 Krylov Bogolyubov Tightness ν s.t. {P t ν : t 0} tight + Feller P t h C b (X) for h C b (X) = existence of an invariant measure Barbu, Da Prato 02 + Tubaro 06 : Wave + additive noise Ichikawa 84 : Take (X, w) instead of (X, ) Maslowski, Seidler 99,01 : Take (X, bw) instead of (X, w) Brzeźniak, Ondreját, Seidler : Wave, beam equation

10 Krylov Bogolyubov Tightness ν s.t. {P t ν : t 0} tight + Feller P t h C b (X) for h C b (X) = existence of an invariant measure Barbu, Da Prato 02 + Tubaro 06 : Wave + additive noise Ichikawa 84 : Take (X, w) instead of (X, ) Maslowski, Seidler 99,01 : Take (X, bw) instead of (X, w) Brzeźniak, Ondreját, Seidler : Wave, beam equation

11 Krylov Bogolyubov Tightness ν s.t. {P t ν : t 0} tight + Feller P t h C b (X) for h C b (X) = existence of an invariant measure Barbu, Da Prato 02 + Tubaro 06 : Wave + additive noise Ichikawa 84 : Take (X, w) instead of (X, ) Maslowski, Seidler 99,01 : Take (X, bw) instead of (X, w) Brzeźniak, Ondreját, Seidler : Wave, beam equation

12 Krylov Bogolyubov Tightness ν s.t. {P t ν : t 0} tight + Feller P t h C b (X) for h C b (X) = existence of an invariant measure Barbu, Da Prato 02 + Tubaro 06 : Wave + additive noise Ichikawa 84 : Take (X, w) instead of (X, ) Maslowski, Seidler 99,01 : Take (X, bw) instead of (X, w) Brzeźniak, Ondreját, Seidler : Wave, beam equation

13 Bounded weak topology Definition V X is bw-closed iff V B r is w-closed r > 0 (X, w) (X, bw) (X, ) Compact (X, ) Compact (X, bw) = Compact (X, w) f : X R is bw-continuous iff x n w x = f (xn ) f (x) Example Closed balls in X are w-compact and bw-compact.

14 Bounded weak topology Definition V X is bw-closed iff V B r is w-closed r > 0 (X, w) (X, bw) (X, ) Compact (X, ) Compact (X, bw) = Compact (X, w) f : X R is bw-continuous iff x n w x = f (xn ) f (x) Example Closed balls in X are w-compact and bw-compact.

15 Bounded weak topology Definition V X is bw-closed iff V B r is w-closed r > 0 (X, w) (X, bw) (X, ) Compact (X, ) Compact (X, bw) = Compact (X, w) f : X R is bw-continuous iff x n w x = f (xn ) f (x) Example Closed balls in X are w-compact and bw-compact.

16 Bounded weak topology Definition V X is bw-closed iff V B r is w-closed r > 0 (X, w) (X, bw) (X, ) Compact (X, ) Compact (X, bw) = Compact (X, w) f : X R is bw-continuous iff x n w x = f (xn ) f (x) Example Closed balls in X are w-compact and bw-compact.

17 Bounded weak topology Definition V X is bw-closed iff V B r is w-closed r > 0 (X, w) (X, bw) (X, ) Compact (X, ) Compact (X, bw) = Compact (X, w) f : X R is bw-continuous iff x n w x = f (xn ) f (x) Example Closed balls in X are w-compact and bw-compact.

18 Krylov Bogolyubov Tightness ν s.t. {P t ν} t 0 tight + Feller P t h C b (X) h C b (X) Weak tightness ν s.t. {Pt ν} t 0 weak tight Bounded weak tightness ν s.t. {Pt ν} t 0 weak tight + + Weak Feller P t h C b (X, w) h C b (X, w) Bounded Weak Feller P t h C b (X, bw) h C b (X, bw)

19 Tightness Classical tighness + parabolic problems on bounded domains - hyperbolic problems - unbounded domains Bounded weak tighness + all problems + all domains = boundedness in probability, i.e. u ε > 0 r > 0 t 0 such that P [ u(t) X > r] < ε

20 Feller property x n x = Eϕ(u xn (t)) Eϕ(u x (t)), ϕ C b (X) Example E u xn (t) u x (t) p κe ct x n x p

21 Weak Feller property x α x = Eϕ(u xα (t)) Eϕ(u x (t)), ϕ C b (X, w) Example Linear equations

22 Bounded weak Feller property x n w x = Eϕ(u x n (t)) Eϕ(u x (t)), ϕ C b (X, w) Example E u xn (t) u x (t) p κe ct x n x p does not help.

23 Bounded weak Feller property du = (Au + f (u)) dt + g(u) dw, x n w x (1) P [sup t T u xn (t) X r] ε (2) f and g bounded on bounded sets (1) and (2) = { Lawu xn } tight (3) tightness on C(R + ; (X, w)) - Polish-like space (4) weak sequential continuity of f and g f (x n ), ϕ f (x), ϕ and ϕ g(x n ) ϕ g(x) 0 (5) Martingale problem: Eϕ(u xn (t)) Eϕ(u x (t))

24 Polish-like spaces Definition (Jakubowski) Y a topological space, ξ n C(Y ) separate points of Y 1. every compact in Y is metrizable 2. a set is compact iff it is sequentially compact 3. every probability measure on a σ-compact is Radon 4. Prokhorov theorem 5. Skorokhod representation theorem 6. regular versions of conditional probabilities

25 Examples of Polish-like spaces Z Polish P(Z ) for Z Polish L (R d ) X separable Fréchet X separable Fréchet with weak topology C(R + ; (X, w)) h t,ϕ = sup { ϕ(h(s)) : s [0, t]}, t 0, ϕ X

26 Examples: Bounded Weak Feller Property Nonlinear wave equation on H 1 loc (Rd ) L 2 loc (Rd ), d 3 u tt = u αu βu t u u p 1 +λ 1 (u) Du +( u q +λ 2 (u) Du) Ẇ for 1 p d+2 d 2, 1 q p+1 2.

27 Examples: Bounded Weak Feller Property Stochastic geometric wave equation on H 2 loc (R) H1 loc (R) u tt = u xx βu t S u (u x, u x ) + S u (u t, u t ) + g u (u t, u x ) Ẇ in a compact Riemannian manifold. u tt = u xx βu t + ( u x 2 u t 2 )u + g u (u t, u x ) Ẇ in Sd 1 Quantum mechanics General relativity Yang-Mills theory Optics

28 Wave equation u tt = u m 2 u au u p 1 βu t + ηg(u) Ẇ on Rd W spatially homogeneous, µ < p [1, ) if d {1, 2} or p [1, 2 d 2 ] if d 3 g(x) 2 c 0 + c 1 x 2 + c 2 x p+1 Theorem If β is large, an invariant measure on W 1,2 (R d ) L 2 (R d ) exists. Proof. Bounded Feller + boundedness in L 0 via Pritchard-Zabczyk.

29 Beam equation Stochastic beam equation on D(A) X u tt + A 2 u + βu t + m( B 1/2 u 2 )Bu = g(u, u t ) Ẇ transversal deflection of an extensible beam under axial force dynamic buckling nonlinear oscillations of a plate in a supersonic flow of gas large amplitude vibrations of an elastic panel excited by aerodynamic forces Theorem If β is large then an invariant measure exists. Proof. Bounded Feller + boundedness in L 0 via Pritchard-Zabczyk.

30 Beam equation - Example 2 ( ) u t 2 m u 2 dx u +γ 2 u +β u D t = Π(x, u, u)q 1/2 Ẇ on D R d with either the clamped boundary condition u = u ν or the hinged boundary condition Q 0 is trace class on L 2 (D). = 0 on D u = u = 0 on D

31 Summary du = (Au + f (u)) dt + g(u) dw (1) weak sequential continuity of f and g (2) f and g bounded on bounded sets (3) P [sup t T u x (t) X r] ε for every x R (4) P [ u(t) X > r] < ε for every t 0 = existence of an invariant measure