Solving Stochastic Partial Differential Equations as Stochastic Differential Equations in Infinite Dimensions - a Review

Transcription

1 Solving Stochastic Partial Differential Equations as Stochastic Differential Equations in Infinite Dimensions - a Review L. Gawarecki Kettering University NSF/CBMS Conference Analysis of Stochastic Partial Differential Equations Based on joint work with V. Mandrekar, B. Rajeev, P. Richard L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

2 Outline 1 SPDE s to Infinite Dimensional SDE s 2 The Infinite Dimensional SDE 3 Existence and Uniqueness Results 4 Strong Solutions from Weak 5 Exponential Ultimate Boundedness L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

3 Infinite Dimensional DE s Two fundamental problems: Peano theorem is invalid in infinite dimansional Banach spaces Theorem (Peano) For each continuous function f : R B B defined on some open set V R B and for each point (t, x ) V the Cauchy problem x (t) = f (t, x(t)), x(t ) = x has a solution which is defined on some neighborhood of t. Theorem (Godunov, 1973) Each Banach space in which Peano s theorem is true is finite dimensional. Appearence of unbounded operators in the equation : W 1,2 W 1,2 (Sobolev space) is unbounded. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

4 SPDE s to Infinite Dimensional SDE s Most popular approaches: Semigroup solution, mild solution to a semiliner DE (Hille, DaPrato, Zabczyk). Solution in a multi-hilbertian space, e.g. in a dual to a nuclear space (Itô, Kallianpur). Variational solution in Gelfand triplet (Agmon, Lions, Röckner). Solutions via Dirichlet forms (Albeverio, Osada (Itô Prize 213)) White noise apporach (Hida) Brownian sheet formulation (Walsh) Solutions in R (Leha, Ritter) L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

5 (S)PDE s to Infinite Dimensional (S)DE s Example (Abstract Cauchy Problem - Semilinear SDE) One dimensional Heat Equation, { ut (t, x) = u xx (t, x), t > u(, x) = ϕ(x) { du(t) dt = u(t), t > u() = ϕ X The Cauchy problem is equation is transformed to an abstract Cauchy problem in the Banach space X of bounded uniformly continuous functions. The differentiation is in the sense of the Banach space. Solution: u(t, x) = (G(t)ϕ) (x). where G(t) is the Gaussian semigroup on the Banach space X 1 exp { x y 2 /4t } ϕ(y) dy, t > (4πt) (G(t)ϕ) (x) = 1/2 R ϕ(x), t =. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

6 SPDE s to Infinite Dimensional SDE s Example (Fréchet nuclear space) Neuronal Models dx t = A X t + B(X t )dw t If ( ) H,, H is a separable Hilbert space and A is an operator with a discrete spectrum, with eigenvalues and eigenvectors λ j > and h j H, such that (1 + λ j ) 2r 1 < for some r1 >, then j=1 Φ = { φ H : } (1 + λ j ) 2r φ, h j 2 H <, r j=1 is a Fréchet nuclear space, with {h j } Φ being a common orthogonal system. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

7 SPDE s to Infinite Dimensional SDE s Let H r, be the completion of Φ with respect to the Hilbertian norms r defined by the inner product f, g r = (1 + λ j ) 2r f, h j H g, h j H, f, g H. j= Φ = r> H r, so that Φ... H r... H H r... Φ For S (R), take A = t 2 d 2 dt 2 I, H = L 2 (R) and h j, Hermite functions. For p R let S p, be the completion of S with respect to the Hilbertian norms p defined by the inner product Then S = p> f, g p = (2k + 1) 2p f, h k L2 g, h k L2, f, g S. k= S p and S = p> S p. A(S) S, A (S ) S. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

8 SPDE s to Infinite Dimensional SDE s Let H r, be the completion of Φ with respect to the Hilbertian norms r defined by the inner product f, g r = (1 + λ j ) 2r f, h j H g, h j H, f, g H. j= Φ = r> H r, so that Φ... H r... H H r... Φ For S (R), take A = t 2 d 2 dt 2 I, H = L 2 (R) and h j, Hermite functions. For p R let S p, be the completion of S with respect to the Hilbertian norms p defined by the inner product Then S = p> f, g p = (2k + 1) 2p f, h k L2 g, h k L2, f, g S. k= S p and S = p> S p. A(S) S, A (S ) S. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

9 SPDE s to Infinite Dimensional SDE s Let H r, be the completion of Φ with respect to the Hilbertian norms r defined by the inner product f, g r = (1 + λ j ) 2r f, h j H g, h j H, f, g H. j= Φ = r> H r, so that Φ... H r... H H r... Φ For S (R), take A = t 2 d 2 dt 2 I, H = L 2 (R) and h j, Hermite functions. For p R let S p, be the completion of S with respect to the Hilbertian norms p defined by the inner product Then S = p> f, g p = (2k + 1) 2p f, h k L2 g, h k L2, f, g S. k= S p and S = p> S p. A(S) S, A (S ) S. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

10 Example (Gelfand Triplet - Variational Solutions) Diffusion Models dx t = AX(t) + B(X t )dw t with A : V = W 1,2 V = W 1,2 and B : V L(R, H) = H = L 2 by Av = α 2 d 2 v dx 2 + β dv + γv + g, v V, dx dv Bv = σ 1 dx + σ 2v, v V. where H = L 2 ((, )), V = W 1,2 ((, )), with the usual norms v H = v V = ( + 1/2, v dx) 2 v H, ( + ( v 2 + ( dv ) 2 ) 1/2, dx) v V. dx L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

11 SPDE s to Infinite Dimensional SDE s Variational Method - Gelfand triplet, V H V, V, H, V are real separable Hilbert spaces, H is identified with its dual H. Embeddings are continuous, dense and compact (or Hilbert-Schmidt) L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

12 Semilinear SDE/SDE if A = dx(t) = (A(X(t)) + F (t, X)) dt + B(t, X) dw t Wiener dx(t) = (A(X(s)) + F (s, X)) ds + B(s, X, u) q(ds, du)poisson X() = ξ F meas. U A : D(A) H H generator of a C semigroup F : [, T ] C([, T ], H) H B : [, T ] C([, T ], H) L 2 (K Q, H)( Wiener) B : [, T ] D([, T ], H) L 2 (U, H)( Poisson) W t is a K valued Q Wiener process, q(ds du) = N(ds du) ds µ(du) is compensated Poisson random measure (cprm). L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

13 Semilinear SDE Mild solution (if A =, strong or weak solution) in C([, T ], H) or D([, T ], H) t X(t) = S(t)ξ + X(t) = S(t)ξ + t t S(t s)f(s, X) dt + S(t s)b(s, X) dw s S(t s)f(s, X) dt t + S(t s)b(s, X, u) q(ds du) U L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

14 Motivation for the Mild Solution Inhomogeneous Initial Value Problem du(t) dt If u is a solution, then and by integrating = Au(t) + f (t), t > u() = x D(A) dt (t s)u(s) ds = T (t s)f (s) t u(t) = T (t)x + T (t s)f (s) ds L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

15 Motivation for the Mild Solution Stochastic Convolution t S(t s)f (s) ds is replaced by S Φ(t) = t S(t s)φ(s) dw s. (see 5.3 D. Khoshnevisan Course notes and Theorem 3.1 in Theorem Assume that A is an infinitesimal generator of a C semigroup of operators S(t) on H, and W t is a K valued Q Wiener process. (a) For h D(A ), then t t X(t), h H = X(s), A h H ds + Φ(s) dw s, h, P a.s., (2.1) iff X(t) = S Φ(t). (b) If Φ Λ 2 (K Q, H), Φ(K Q ) D(A), and AΦ Λ 2 (K Q, H), then S Φ(t) is a strong solution. H L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

16 Multi Hilbertian (Fréchet nuclear) space SDE dx(t) = F(t, X(t))) dt + B(t, X(t)) dw t Wiener t t X(t) = F (s, X(s))) ds + B(s, X(s ), u) q(ds, du)poisson X() = ξ F meas. U F : [, T ] Φ Φ B : [, T ] Φ L(Φ, Φ )( Wiener) B : [, T ] Φ U Φ ( Poisson) Solution, continuous or cadlag, is Φ valued, but found in H p, some p >. φ, X(t) = φ, ξ + φ, X(t) = φ, ξ + t t t φ, F(s, X(s)) dt + φ, B(s, X(s)) dw s t φ, F(s, X) dt + B(s, X(s ), u)[φ] q(du ds) L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

17 Variational Method Variational SDE dx(t) = A(t, X(t))dt + B(t, X(t)) dw t with the coefficients A : [, T ] V V and B : [, T ] V L 2 (K Q, H) and an H valued F measurable initial condition ξ L 2 (Ω, H). (W) X(t) = ξ + (P) X(t) = ξ + P a.s t t A(s, X(s)) ds + A(s, X(s)) ds + t t B(s, X(s)) dw s, P a.s. U B(s, X(s ), u) q(ds du), The integrants A and B are evaluated at a V valued F t measurable version of X(t) in L 2 ([, T ] Ω, V ). L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

18 Solving the Equation A, B Martingale Rep. Thm. X weak soln. in Characterize the Lim. C or D([, T ], V ) V H V a n, b n X n weak soln. in x R n, C or D([, T ], R n ) n a n (t, x) j = φ i, A(t, x k φ k j k=1 R n R n R n a n,l, b n,l, l Lip. Approx. X n,l strong soln. in L 2 E sup t X n,l,k (t) X n,l (t) 2 R n Pickard X n,l,k L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

19 Semilinear SDE/SDE when A = - Coefficients (M) F and B are jointly measurable, and for every t T, they are measurable with respect to the product σ field F t C t on Ω C([, T ], H), where C t is a σ field generated by cylinders with bases over [, t]. (JC) F and B are jointly continuous. (G-F-B) There exists a constant l, such that x C([, T ], H) ( ) F (ω, t, x) H + B(ω, t, x) L2 (K Q,H) l 1 + sup x(s) H, s T for ω Ω, t T. (A4) For all x, y C([, T ], H), ω Ω, t T, there exists K >, such that F(ω, t, x) F(ω, t, y) H + B(ω, t, x) B(ω, t, y) L2 (K Q,H) K sup x(s) y(s) H. s T L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

20 Infinite Dimensional SDE (A = ) - Solutions Lipschitz case - strong solutions exist and are unique (Pickard) Continuous case - Lipschitz Approximation F n (t, x) = F (t, (γ n (, x,..., x n ), e)) smoothing kernel Existence result for dx(t) = F(t, X)) dt + B(t, X) dw t Theorem Let H 1 be a real separable Hilbert space. Let the coefficients F, B of the SDE satisfy conditions (M), (JC), (G-F-B) on H 1. Assume that there exists a Hilbert space H such that the embedding J : H H 1 is a compact operator (failure of the Peano theorem) and that F, B restricted to H satisfy F : [, T ] C([, T ], H) H, B : [, T ] C([, T ], H) L(K, H), and the linear growth condition (G-F-B). Then the SDE has a weak solution X( ) C([, T ], H 1 ). L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

21 Smoothing Kernel Let {e n } n=1 be an ONB in H. Denote f n (t) = ( x(t), e 1 H, x(t), e 2 H,..., x(t), e n H ) R n, Γ n (t) = f n (kt /n) at t = kt /n and linear otherwise, γ n (t, x,..., x n ) = x k at t = kt n and linear otherwise, with x k R n, k =, 1,..., n. Let g : R n R be non-negative, vanishing for x > 1, possessing bounded derivative, and such that g(x)dx = 1. Let ε n. We define R n F n (t, x) = { F (t, (γ n (, x,}..., x n ), e)) exp ε n n ( ( n xk 2 fn ( kt n g t) x ) k n k= k= ε n dx k ε n ) (3.1) Above, (γ n (, x,..., x n ), e) = γ 1 ne γ n ne n, where γ 1 n,..., γ n n are the coordinates of the vector γ n in R n, and x 2 k = n i=1 (x i k )2, dx k = dx 1 k... dx n k. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

22 Semilinear SDE (A ) - Solutions Lipschitz case - strong solutions exist and are unique Continuous case - Lipschitz Approximation Theorem Assume that A is an infinitesimal generator of a compact C semigroup S(t) (Peano) on a real separable Hilbert space H. Let the coefficients of the Semilinear SDE satisfy conditions (M), (JC), (G-F-B). Then the Semilinear SDE dx(t) = (AX(t) + F (t, X)) dt + B(t, X) dw t has a martingale solution (i.e. weak mild solution). L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

23 Multi-Hilbertian Space; Coefficients are Differential Operators. Consider an SDE dx t = L(X t )dt + A(X t )db t (3.2) For 1 i d, let i : S S be the differentiation operators. Then i extends in the usual manner as an operator i : S S. Let i denote the transpose of i. Then i : S S is given by i u = i u, u S. Define A : S L ( R d, S ) and L : S S by Au(x) = Lu = 1 2 with u S, x = (x 1 x d ) R d. d ( i u)x i i=1 d i 2 u, i=1 L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

24 Multi-Hilbertian Space; Coefficients are Differential Operators. Consider an SDE dx t = L(X t )dt + A(X t )db t (3.2) For 1 i d, let i : S S be the differentiation operators. Then i extends in the usual manner as an operator i : S S. Let i denote the transpose of i. Then i : S S is given by i u = i u, u S. Define A : S L ( R d, S ) and L : S S by Au(x) = Lu = 1 2 with u S, x = (x 1 x d ) R d. d ( i u)x i i=1 d i 2 u, i=1 L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

25 Multi-Hilbertian Space; Coefficients are Differential Operators. Consider an SDE dx t = L(X t )dt + A(X t )db t (3.2) For 1 i d, let i : S S be the differentiation operators. Then i extends in the usual manner as an operator i : S S. Let i denote the transpose of i. Then i : S S is given by i u = i u, u S. Define A : S L ( R d, S ) and L : S S by Au(x) = Lu = 1 2 with u S, x = (x 1 x d ) R d. d ( i u)x i i=1 d i 2 u, i=1 L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

26 Growth Properties of the Coefficients For a bounded linear operator T L ( ) R d, H p, its Hilbert Schmidt norm is calculated as T HS(p) = ( d 1/2, i=1 Te i p) 2 where {ei } d i=1 is the canonical basis in R d. Proposition For the differential operators i, A, and L defined above the following properties hold true: (a) For any p q + 1/2, and 1 i d, i : S p S q is continuous, and for u S p, i u q C q u p, where the constant C q depends (only) on q. (b) For any p q + 1, and u S p, Lu q D q u p Au HS(q) D q u p, where the constant D q depends (only) on q. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

27 Growth Properties of the Coefficients For a bounded linear operator T L ( ) R d, H p, its Hilbert Schmidt norm is calculated as T HS(p) = ( d 1/2, i=1 Te i p) 2 where {ei } d i=1 is the canonical basis in R d. Proposition For the differential operators i, A, and L defined above the following properties hold true: (a) For any p q + 1/2, and 1 i d, i : S p S q is continuous, and for u S p, i u q C q u p, where the constant C q depends (only) on q. (b) For any p q + 1, and u S p, Lu q D q u p Au HS(q) D q u p, where the constant D q depends (only) on q. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

28 Growth Properties of the Coefficients For a bounded linear operator T L ( ) R d, H p, its Hilbert Schmidt norm is calculated as T HS(p) = ( d 1/2, i=1 Te i p) 2 where {ei } d i=1 is the canonical basis in R d. Proposition For the differential operators i, A, and L defined above the following properties hold true: (a) For any p q + 1/2, and 1 i d, i : S p S q is continuous, and for u S p, i u q C q u p, where the constant C q depends (only) on q. (b) For any p q + 1, and u S p, Lu q D q u p Au HS(q) D q u p, where the constant D q depends (only) on q. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

29 Source of the Main Technical Problem Why approximate solutions travel from space to space Proof. Part (b) follows from (a). Since i i u 2 q = = = i, for u S p, we have (2 k + d) 2q i u, h k 2 k = (2 k + d) 2q u, i h k 2 k = 2 2q (2 k + d) 2(q+ 1 2) u, hk 2 C q u 2 p. k = using the recurrence relation l l + 1 h l (x) = 2 h l 1(x) 2 h l+1(x). L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

30 The Monotonicity Condition What keeps things within one space Theorem 2 L(u v), u v p + Au Av 2 HS(p) θ u v 2 p, holds true for r p + 1, u, v S r. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

31 Equation Coefficients are Differential Operators Operators A(u)(h) = L(u) = 1 2 d ( i u)h i (A = ) i=1 d i 2 u (L = 1 2 ) i=1 u S, h R d, satisfy our conditions with q p + 1. The unique solution of { dxt = 1 2 X tdt X t db t X = φ S p is φ( + B t ). If φ = δ, then X t = δ Bt. Monotonicity holds true for more general differential operators L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

32 Equation Coefficients are Differential Operators Operators A(u)(h) = L(u) = 1 2 d ( i u)h i (A = ) i=1 d i 2 u (L = 1 2 ) i=1 u S, h R d, satisfy our conditions with q p + 1. The unique solution of { dxt = 1 2 X tdt X t db t X = φ S p is φ( + B t ). If φ = δ, then X t = δ Bt. Monotonicity holds true for more general differential operators L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

33 Equation Coefficients are Differential Operators Operators A(u)(h) = L(u) = 1 2 d ( i u)h i (A = ) i=1 d i 2 u (L = 1 2 ) i=1 u S, h R d, satisfy our conditions with q p + 1. The unique solution of { dxt = 1 2 X tdt X t db t X = φ S p is φ( + B t ). If φ = δ, then X t = δ Bt. Monotonicity holds true for more general differential operators L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

34 Variational Method On a tripple V H V This set up arises in the study of SPDE s. Typical example is H = L 2 (O), V = W 1,2 Sobolev space Together with other regularity assumptions, the following coercivity condition is imposed 2 Lu, u + Au 2 HS(H) δ u 2 V + η u 2 H This condition is violated in our case of differential operators! Let X = S 1, X = S 2 1, H = L 2. Then (X, H, X ) is a normal triple with 2 canonical bilinear form given by the L 2 inner product. Then for ξ S X, 2 ξ, Lξ + Aξ 2 HS() + δ ξ 2 1 = δ ξ which cannot be dominated by using the L 2 norm. Note that the equality 2 ξ, Lξ = Aξ 2 HS() follows from integration by parts. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

35 Variational Method On a tripple V H V This set up arises in the study of SPDE s. Typical example is H = L 2 (O), V = W 1,2 Sobolev space Together with other regularity assumptions, the following coercivity condition is imposed 2 Lu, u + Au 2 HS(H) δ u 2 V + η u 2 H This condition is violated in our case of differential operators! Let X = S 1, X = S 2 1, H = L 2. Then (X, H, X ) is a normal triple with 2 canonical bilinear form given by the L 2 inner product. Then for ξ S X, 2 ξ, Lξ + Aξ 2 HS() + δ ξ 2 1 = δ ξ which cannot be dominated by using the L 2 norm. Note that the equality 2 ξ, Lξ = Aξ 2 HS() follows from integration by parts. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

36 Existence of Weak Variational Solutions Theorem Let V H V be a Gelfand triplet (Unbounded Operator)with compact inclusions. Let the coefficients A, B of Equation dx(t) = A(t, X(t))dt + B(t, X(t)) dw t satisfy conditions [JC], [G-A], [G-B], and [C]. Let the initial condition ξ be an H valued random variable satisfying [IC]. Then there exists a weak solution X(t) in C([, T ], H), such that ( ) E sup X(t) 2 H t T <, and E T X(t) 2 V dt <. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

37 Conditions (JC) (Joint Continuity) The mappings are continuous (t, v) A(t, v) V (t, v) B(t, v)qb (t, v) L 1 (H) For some constant θ, (G-A) (Growth on A - (Unbounded Operator)) A(t, v) 2 V θ ( 1 + v 2 H), v V. (G-B) (Growth on B) B(t, v) 2 L 2 (K Q,H) θ ( 1 + v 2 H), v V. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

38 Coercivity condition on A and B Initial condition (C) There exist constants α >, γ, λ R such that for v V, 2 A(t, v), v + B(t, v) 2 L 2 (K Q,H) λ v 2 H α v 2 V + γ. (IC) For some constant c. { ( ( )) } 2 E ξ 2 H ln 3 + ξ 2 H < c, L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

39 Existence for Wiener noise (cprm - similar) Result on the existence of a weak solution. Theorem Let Bt n be a standard Brownian motion in R n. There exists a weak solution to the following finite dimensional SDE, dx(t) = a(t, X(t))dt + b(t, X(t)) db n t, with an R n valued F measurable initial condition ξ n, if a : [, ] Rn R n, b : [, ] R n R n R n are continuous and satisfy the following growth condition b(t, x) 2 L(R n ) K ( 1 + x 2 R n ) x, a(t, x) R n K ( 1 + x 2 R n ) for t and x R n and some constant K. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

40 Approximation Problem Lemma The growth conditions [G-A] and [G-B] assumed for the coefficients A and B imply the following growth conditions on a n and b n, (Unbounded Operator) a n (t, x) 2 R n θ n ( 1 + x 2 R n), θn tr (σ n (t, x)) = tr ( b n (t, x) (b n (t, x)) T ) θ ( 1 + x 2 R n ). The coercivity condition [C] implies that for a large enough value of θ, ( ) 2 a n (t, x), x R n + tr b n (t, x) (b n (t, x)) T θ ( 1 + x 2 ) R. n The constant θ n depends on n, but θ does not. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

41 Tightness in C([, T ], V ) - similar in D([, T ], V ). Theorem Let the coefficients A, B of Equation dx(t) = A(t, X(t))dt + B(t, X(t)) dw t satisfy conditions [JC], [G-A], [G-B], and [C]. Consider the family of measures µ n on C([, T ], V ), with support in C([, T ], H), defined by } n µ n (Y ) = µ {x n C([, T ], R n ) : x i (t)ϕ i Y ; Y C([, T ], V ), where µ n are distributions of finite dimensional solutions, ϕ i, i = 1,... is an ONB in H, consisting of elements from V. If the embedding H V is compact (H-S in D). Then the family of measures {µ n } n=1 is tight on C([, T ], V ) (D([, T ], V )). i=1 L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

42 Tightness in C([, T ], V ) and D([, T ], V ) Theorem (Mitoma: C([, T ], S ), D([, T ], S )). Let V H V be Gelfand triplet with Hilbert Schmidt embeddings. Given {µ n } n=1 Borel probability measures on C([, T ], V ) (D([, T ], V ) ), s.t. 1 {µ n π 1 is tight on C([, T ], R) (D([, T ], R) ) 2 ε M n V } n=1 µ n {f C([, T ], V ) : sup f (t) H > M} < ε t (µ n {f D([, T ], V ) : sup f (t) H > M} < ε) t then {µ n } n=1 is tight on C([, T ], V ) (D([, T ], V ) ). Thus a n H is not a problem as we take one dimensional projections only, but the price is H S embedding. This can be improved in C([, T ], V ). L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

43 Coming back from V to H By the Skorokhod theorem, X n X a.s. in C([, T ], V ). Consider α H : V R, α H (u) = sup { u, v, v V, v H 1}. α H (u) = u H if u H, and is lower semicontinuous as a sup of continuous functions. For u V \ H, α H (u) = + Thus, we can extend the norm H to a lower semicontinuous function on V. By the Fatou lemma, ( ) sup x(t) 2 H µ (dx) = E sup X(t) 2 H C([,T ],V ) t T t T ( ) E lim inf n lim inf n E = lim inf n ( sup t T X n (t) 2 H sup X n (t) 2 H t T sup C([,T ],V ) t T ) x(t) 2 H µ n (dx) < C. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

44 Coming back from H to V Apply the Itô formula and Coercivity E X n (t) 2 H = E ξ n 2 H + 2E +E t t a n (s, X n (s)), X n (s) R n ds ( ) tr b n (s, X n (s)) (b n (s, X n )) T ds t t E ξ 2 H + λ E X n (s) 2 H ds α E X n (s) 2 V ds + γ. Conclude that T sup E X n (t) 2 V dt <. n Extend the norm V to a lower semicontinuous function on V α V (u) = sup { u, v, v V, v V 1}, since α V (u) = u V if u V, and for u V \ V, α V (u) = +. By the Fatou lemma T x(t) 2 V dt µ (dx) <. C([,T ],V ) L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

45 Characterization of the limit t M t (x) = x(t) x() A(s, x(s)) ds, is, in either case (Wiener, cprm), a martingale. Three steps: Proving that M t is a martingale by evaluating ( M t (x) M s (x), v g s (x)) µ (dx) = for a bounded function g s on C([, T ], V ), which is measurable with respect to the cylindrical σ field generated by the cylinders with bases over [, s], Finding its increasing process < M > t Using Martingale Representation Theorem First two steps use uniform integrability. Wiener - usually of X 2 n (t), cprm - more delicate. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

46 Lions Theorem (extension of)- H continuous version: Theorem Let X() L 2 (Ω, H), Y L 2 ([, T ] Ω, V ), Z L 2 ([, T ] Ω, L 2 (K Q, H)) be both progressively measurable. Define a continuous V valued process t t X(t) = X() + Y (s) ds + Z (s) dw s, t [, T ]. If for its dt P equivalence class ˆX we have ˆX L 2 ([, T ] Ω, V ), then X is an H valued continuous F t adapted process, E sup X(t) 2 H < t [,T ] L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

47 Pathwise Uniqueness Definition If for any two H valued weak solutions (X 1, W ) and (X 2, W ) of Equation dx(t) = A(t, X(t))dt + B(t, X(t)) dw t defined on the same filtered probability space (Ω, F, {F t } t T, P) and with the same Q Wiener process W, such that X 1 () = X 2 (), P a.s., we have that P (X 1 (t) = X 2 (t), t T ) = 1, then we say that this Equation has pathwise uniqueness property. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

48 The weak monotonicity condition (WM) There exists c R, such that for all u, v V, t [, T ], 2 u v, A(t, u) A(t, v) + B(t, u) B(t, v) 2 L 2 (K Q,H) c u v 2 H. Weak monotonicity is crucial in proving uniqueness of weak and strong solutions. In addition, it allows to construct strong solutions in the absence of the compact embedding V H. Theorem Let the conditions [JC], [GB], [C], [IC] hold true and assume the weak monotonicity condition [WM] and (G-A) (Growth on A) Then the solution to the variational SDE A(t, v) 2 V θ ( 1 + v 2 V ), v V. dx(t) = A(t, X(t))dt + B(t, X(t)) dw t is pathwise unique. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

49 Proof. Let X 1, X 2 be two weak solutions, Y (t) = X 1 (t) X 2 (t), and denote its V valued progressively measurable version by Ȳ. Applying the Itô formula and the monotonicity condition [WM] yields t e θt Y (t) 2 H = θ t e θs Y (s) 2 H ds + e θs( 2 Ȳ (s), A(s, X 1 (s)) A(s, X 2 (s)) ) + B(s, X 1 (s)) B(s, X 2 (s)) 2 L 2 (K Q,H) ds +2 M t, t e θs Y s, (B(s, X 1 (s)) B(s, X 2 (s))) dw s H where M t is a real valued continuous local martingale represented by the stochastic integral above. The inequality above also shows that M t. Hence by the Doob maximal inequality, M t =. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

50 As a consequence of an infinite dimensional version of the result of Yamada and Watanabe we have the following corollary. Corollary Under conditions of the Existence Theorem and assuming [WM] (weak monotonicity), the variational SDE dx(t) = A(t, X(t))dt + B(t, X(t)) dw t has unique strong solution. Yamada-Watanabe argument does not go through in general for cprm, but works if U is separable. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

51 Exponential Ultimate Boundedness Definition We say that the variational solution of the variational SDE dx(t) = A(t, X(t))dt + B(t, X(t)) dw t is exponentially ultimately bounded in the mean square sense (m.s.s.), if there exist positive constants c, β, M, such that E X x (t) 2 H c e βt x 2 H + M, for all x H. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

52 Theorem The strong solution {X x (t), t } of equation dx(t) = A (X(t)) dt + B (X(t)) dw t X() = x H where A and B are in general non linear mappings, is exponentially ultimately bounded in the m.s.s. if there exists a function Ψ : H R to which Itô s formula can be applied and, in addition, such that (1) c 1 x 2 H k 1 Ψ(x) c 2 x 2 H + k 2, for some positive constants c1, c 2, k 1, k 2 and for all x H, (2) LΨ(x) c 3 Ψ(x) + k 3, for some positive constants c 3, k 3 and for all x V. where ( ) LΨ(u) = Ψ (u), A(u) + tr Ψ (u)b(u)qb (u). L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

53 If A and B are linear and satisfy coercivity condition, the Lyapunov function can be written explicitly Ψ (x) = T t E X x (s) 2 V ds dt, for T large enough. L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

54 Our example SPDE, ( d t u(t, x) = α 2 2 u(t, x) x 2 + β + u(t, x) x ( σ 1 u(t, x) x + σ 2 u(t, x) ) + γu(t, x) + g(x) ) dw t, u(, x) = ϕ(x) L 2 ((, )) L 1 ((, + )), If 2α 2 + σ1 2 <, then the coercivity and weak monotonicity conditions hold true. The growth [G-B] holds and A(t, v) 2 V θ ( 1 + v 2 V ), v V. and that there exists a unique strong solution u ϕ (t) in L 2 (Ω, C([, T ], H)) L 2 (Ω [, T ], V ). Then we can conclude that the solution is exponentially ultimately bounded in the m.s.s. by reducing the case to a liner equation (dropping g). dt L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46

55 L. Gawarecki, V. Mandrekar. Stochastic Differential Equations in Infinite Dimensions with Applications to Stochastic Partial Differential Equations. Springer (211). L. Gawarecki ( Kettering University ) Solutions to Infinitely Dimensional SDE s August 19-23, / 46