Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term

Transcription

1 1 Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term Enrico Priola Torino (Italy) Joint work with G. Da Prato, F. Flandoli and M. Röckner Stochastic Processes and Differential Equations in Infinite Dimensional Spaces, London, 3 March - 4 April 214

2 Introduction We study pathwise uniqueness for the following SDE in a real separable Hilbert space H (cf. [DaPratoZabczyk92]) or (H1) A : D(A) H H, dx t = (AX t + B(X t ))dt + dw t, X = x H (1) X t = e ta x + e (t s)a B (X s ) ds + e (t s)a dw s, t, Ae k = λ k e k where (e k ) D(A) is an orthonormal basis in H, each λ k is positive and, for some δ (, 1), 1 <, k 1 λ 1 δ k (H2) W = (W t ) = (W t ) t is a cylindrical Wiener process in H. Formally, W t = + k=1 β k (t)e k, (H3) B : H H is Borel measurable and locally bounded (i.e., bounded on balls). 2

3 3 This is a continuation of - Da Prato, G., Flandoli, F., P., Röckner, M.: Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift, Annals of Probability 212. More precisely, we will localize our previous result : Theorem (Da Prato, Flandoli, P., Röckner, AOP 212) Assume (H1), (H2) and suppose that B : H H is Borel and bounded. Let µ be the centered Gaussian measure on H with covariance Q = 1 2 A 1. Then, for µ-a.e. (deterministic) x H, there is a unique (in the pathwise sense) mild solution of (1). ote that µ := (, Q), Q = 1 2 A 1. is the unique invariant measure for the associated Ornstein-Uhlenbeck process corresponding to B =.

4 4 The theorem gives a generalization of a result by Veretennikov (198) proved when H = R d. The result is due to the non-degeneracy of the noise W in an essential way. In the proof we have used An Itô-Tanaka trick which is based on regularity results for infinite dimensional elliptic equations in Sobolev spaces with respect to µ (see Chojnowska-Michalik and Goldys, J. Funct. Anal. 21 and Probab. Theory Relat. Fields 22). The Girsanov theorem. An example. Consider the following semilinear parabolic SPDE in D = [, π] (H = L 2 (D)): dx(t, ξ) = [ 2 X(t, ξ) + B(X(t, ))(ξ)]dt + dw ξξ t(ξ), X(, ξ) = x(ξ), ξ D, (2) X(t, ξ) =, ξ D, t >

5 Basic definitions 5 A : D(A) H H is a negative definite self-adjoint operator and ( A) 1+δ, for some δ (, 1), is of trace class. Since A 1 is compact, there exists an orthonormal eigenbasis (e k ) in H and positive eigenvalues (λ k ) with Ae k = λ k e k. We need k 1 In the previous example λ k = k 2. 1 λ 1 δ k <. Recall that A generates an analytic semigroup e ta on H such that, for t, e ta e k = e λ kt e k. We consider a cylindrical Wiener process W = (W t ) with respect to (e k ) which is formally given by W t = β k (t)e k k 1 where β k (t) are independent one dimensional Wiener process defined on the same filtered probability space (Ω, F, (F t ), P) (see [Da Prato-Zabczyk, 1992]).

6 6 We recall the notions of mild solutions to (1) which we use. For more details, we refer to Ondreját, M., Dissertationes Math. 24, and Liu, W., Röckner, M., Introduction to Stochastic Partial Differential Equations. Definition Assume (H1), (H2) and (H3) and let x H. (a) We call weak mild solution to (1) a tuple (Ω, F, (F t ), P, W, X), where (Ω, F, (F t ), P) is a filtered probability space on which it is defined a cylindrical (F t )-Wiener process W and a continuous (F t )-adapted H-valued process such that, P-a.s., X = (X t ) = (X t ) t X t = e ta x + e (t s)a B (X s ) ds + e (t s)a dw s, t. (3) (b) A weak mild solution X which is ( F W t )-adapted (here ( F W t ) denotes the completed natural filtration of the cylindrical process W) is called strong mild solution.

7 7 ote that e (t s)a dw s = n 1 e (t s)λn e n dβ n (s) and the series converges in L 2 (Ω, P; H), for any t. Existence of weak mild solutions on some filtered probability space is known if the measurable drift B grows at most linearly (see [DaPrato-Zabczyk, 1992] which uses the Girsanov theorem). When B = our assumption on A guarantees that the OU process Z t = Z(t, x) = e ta x + e (t s)a dw s has a continuous H-valued version. The Gaussian law of Z(t, x) is (e ta x, Q t ) with mean e ta x and covariance operator Q t, Q t = 1 2 A 1 (I e 2tA ), t. Moreover µ = (, 1 2 A 1 ) is the unique invariant measure.

8 Statement of the local pathwise uniqueness result 8 We consider the first exit time from B(, ) (here B(, ) is the open ball with center and radius 1) (τ X = + if the set is empty). τ X = inf{t : X t B(, )} and also two weak mild solutions X and Y, having the same initial condition x H, and solving the same equation with the same cylindrical Wiener process W, but with possibly different drift terms, respectively B and B B b,loc (H, H), i.e., dx t = (AX t + B(X t ))dt + dw t, X = x, (4) dy t = (AY t + B (Y t ))dt + dw t, Y = x. (5)

9 9 Theorem (Da Prato, Flandoli, P., Röckner to appear in J. Th. Prob.) Assume (H1), (H2) and (H3), so we are assuming B B b,loc (H, H). Then for µ-a.e. x H, if X and Y are two weak mild solutions, respectively of (4) and (5), defined on the same filtered probability space (Ω, F, (F t ), P) with the same cylindrical Wiener process W, and if, for some 1, B(x) = B (x), x B(, ), (6) then, P-a.s., and so τ X = τy, P-a.s.. X t τ X = Y τy t τ X, t, (7) τy Remark If B = B the result implies that, P-a.s., X t = Y t, t (one uses that τ X + and τy + as, since X and Y are both global solutions).

10 Related references on pathwise uniqueness for SDEs and SPDEs (which fails without the noise) - A.J. Veretennikov : Mat. Sb., (.S.) (198) [ B L (R d ) (for any d 1) ]. He extends A. K. Zvonkin : Mat. Sb. (.S.) (1974) [ B L (R), i.e., d = 1 ] I. Gyöngy and E. Pardoux, On the regularization effect of space-time white noise on quasi-linear parabolic PDEs, Probab. Theory Relat. Fields I. Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations, Stochastic Process. Appl V. Krylov and M. Röckner : Probab. Theory Relat. Fields (25) [B L p loc (Rd ) when p > d, generalizing Portenko(1982)] - Davie A. M.: Uniqueness of solutions of stochatic differential equations, Int. Math. Res. otices IMR (27) - F. Flandoli, M. Gubinelli and E. P. : Invent. Math. (21) [ B C α b ; uniqueness for stochastic transport equation]. - G. Da Prato and F. Flandoli: J. Funct. Anal. (21) [pathwise uniqueness for SDEs in Hilbert spaces with B C α (H, H) ]. b - F. Flandoli, Random perturbation of PDEs and fluid dynamic models, LM S. Cerrai, G. Da Prato, F. Flandoli, Pathwise uniqueness for stochastic reaction-diffusion equations in Banach spaces with an Holder drift component, Stochastic PDEs (213). 1

11 11 A tool: L p (H, µ)-regularity results The OU operator is defined on smooth functions ϕ as Lϕ(x) = 1 2 Tr(D2 ϕ(x)) + Ax, Dϕ(x), where Dϕ(x) and D 2 ϕ(x) denote first and second Fréchet derivatives. R t can be extended to a unique C -semigroup on L p (H, µ), p 1. Let L p be its generator. Recall that W 2,p (H, µ), p 1, is defined as the completion of a suitable set of smooth functions endowed with the Sobolev norm. Theorem (Chojnowska-Michalik, B. Goldys, 21) Let λ >, f L p (H, µ), p > 1, and let ϕ D(L p ) be the solution of the equation λϕ L p ϕ = f. (8) Then ϕ W 2,p (H, µ), ( A) 1/2 Dϕ L p (H, µ; H) and C = C(λ, p) such that ϕ L p (µ) + ( D 2 ϕ(x) p µ(dx) ) 1/p + ( A) 1/2 Dϕ L p (µ) C f L p (µ). H ( denotes the Hilbert-Schmidt norm).

12 12 Using the previous result we can study the elliptic equation where λ > is given, f B b (H) and B B b (H, H). Proposition (DFPR, AOP12) λu L 2 u B, Du = f, (9) Let λ λ, where λ := 4C B 2. Then there is a unique solution u D(L 2) of (9) given by u = u λ = (λ L 2 ) 1 (I T λ ) 1 f, where Moreover, u C 1 (H) with b T λ ϕ := B, D(λ L 2 ) 1 ϕ. u 2 f, Du 2C 1 f, and, for any p 2, u W 2,p (H, µ) and, for some C = C(λ, p, B ), D 2 u(x) p µ(dx) C H λ 1 2 f (x) p µ(dx). (1) H

13 A key-point in the proof of the local uniqueness: a local Itô-Tanaka-type formula for solutions A simple Itô-Tanaka-type formula. To simplify we consider H = R and A =. We write ow if v is a regular solution of X t x W t = B(X s )ds. λv Lv = B on R, λ >, L = 1 d + B(x) then by Itô s formula: 2 dx 2 dx v(x t ) = v(x) + v (X s )dw s + Lv(X s )ds and so and d 2 v(x t ) = v(x) + v (X s )dw s + (λv(x s ) B(X s ))ds X t + v(x t ) = x + v(x) + W t + v (X s )dw s + λ v(x s )ds. 13

14 14 A remark on the Zvonkin method To simplify we consider H = R and A =. We have ow if g(t, x) is a regular solution of then by Itô s formula: X t = x + W t + B(X s )ds. t g + Lg = on [, T] R, g(t, x) = x T X T = g(, x) + x g(s, X s )dw s

15 Coming back to our SPDE 15 Recall the notation B = B 1 B(,), 1, (11) where B(, ) is the open ball of radius (hence B is bounded). For any i we denote the i th component of B by B (i), i.e., B (i) (x) := B(x), e i, x H. Then for λ λ we consider the solution u (i) of the equation λu (i) Lu(i) B, Du (i) = B(i), µ -a.e. (12) The next result is a kind of local version of a correponding result in [DFPR12].

16 16 Theorem Let X = (X t ) be a weak mild solution. Consider the stopping time τ X = inf{t : X t B(, )} Let u (i) be the solution of (12) and set X(i) t = X t, e i. For any t > we have P-a.s. on the event {t τ X} Proof. X (i) t I Step. It is enough to prove τ X = λ u (i) (X s)ds = e λit ( x, e i + u (i) (x)) u(i) (X t) + (λ + λ i ) e λi(t s) u (i) s)ds + e λi(t s) (d W s, e i + Du (i) s), dw s ). (13) u (i) (X t τ X τ X ) u (i) (x) (14) B (i) (X s )ds + τ X Du (i) (X s), dw s.

17 Indeed using the equation dx (i) t = λ i X (i) t dt + B (i) (X t )dt + dw (i) t. and inserting the expression for B (i) (X t ), we obtain = X i t τ X τ X + x i + λ u (i) u (i) (X t τ X ) u (i) (x) (X s)ds λ i τx +W i. t τ X X (i) s ds + τ X Du (i) (X s), dw s By the variation of constants formula this gives the assertion on {t τ X }. II Step. Let B,n of C class with all its derivatives which are bounded. Moreover, B,n B, n 1, and ow we denote by u (i) the solution of,n B,n B, µ a.e.. (15) λu (i),n Lu(i),n B,n, Du (i),n = B(i),n (16) where B (i),n = B,n, e i. 17

18 18 We have, possibly passing to a subsequence, for any x H, lim n u(i) (x) = u(i) (x),,n where u (i) is the solution of (12). The idea behind (17) is that if (f n ) satisfies lim n Du(i),n (x) = Du(i) (x), (17) sup u (i),n C 1 b n 1 (H) = C i, <, f n (x) f (x), µ-a.e., f n M. then, for any x H (not only µ-a.e.), t >, R t f n (x) R t f (x) (18) as n, since the law of the OU process Z(t, x) at time t > is absolutely continuous with respect to µ.

19 19 III Step. By using finite dimensional approximation of the solution X t we get for r ], t] and so u (i) (X,n t τ X ) u (i) (X,n r τ X ) τ X ( = Lu (i) (X r τ X,n s) + B,n (X s ), Du (i) (X,n s) ) ds τ X + Du (i) (X r τ X,n s), (B(X s ) B,n (X s )) ds + u (i) (X,n t τ X ) u (i) τ X = λ u (i) (X r τ X,n s)ds τ X r τ X Du (i),n (X s), dw s.,n (X r τ X ) (19) τ X r τ X τ X + Du (i) (X r τ X,n s), (B(X s ) B,n (X s )) ds + B (i),n (X s)ds τ X r τ X ow (19) implies (14) if we can pass to the limit as n. Du (i),n (X s), dw s.

20 2 The difficult term which must go to as n is τ X r τ X Du (i),n (X s), (B(X s ) B,n (X s )) ds = J n + I n, τ X where J n = Du (i) (X r τ X,n s) Du (i) (X s), (B(X s ) B,n (X s )) ds, τ X I n = Du (i) (X r τ X s), (B (X s ) B,n (X s )) ds (using that s t τ X ). As for J n we have τ X J n = Du (i) (X r τ X,n s) Du (i) (X s), (B (X s ) B,n (X s )) ds, and so it tends to, P-a.s., as n.

21 21 Let us consider I n, τ X I n = Du (i) (X r τ X s), (B (X s ) B,n (X s )) ds. We define an auxiliary process ˆX = ( ˆX t ) as follows: ˆX t := e ta x + e (t s)a B (X s τ X )ds + e (t s)a dw s, t, (2) ote that X s τ X = ˆX, s, so that s τ X I n = τ X r τ X Du (i) ( ˆX s ), (B ( ˆX s ) B,n ( ˆX s )) ds D (i) u B ( ˆX s ) B,n ( ˆX s ) ds. (21) r ow thanks to the auxiliary process we can use the Girsanov theorem.

22 22 Let T >. Since where ˆB s that ˆX t := e ta x + e (t s)a ˆB s ds + e (t s)a dw s, t, = B ( ˆX s τ X ), s, is an adapted and bounded process, we have W t := W t + ˆB s ds is a cylindrical Wiener process on ( Ω, F, (F t ) t [,T], P ) where d P dp Hence ˆX t ρ = exp ( T ˆB s dw s 1 2 T ˆB s 2 ds ). = ρ, FT = e ta x + e(t s)a d W s is an OU process on (Ω, F, (F t ) t [,T], P ). Moreover the law of ( ˆX t ) t [,T] on C([, T]; H) (under P) is equivalent to the law of the OU process Z(t, x). In particular, all their transition probabilities are equivalent.

23 23 Since the law of Z(t, x) is equivalent to µ for all t > and x H, we obtain that the law π t (x, ) of ˆX t is absolutely continuous with respect to µ. It follows E B ( ˆX s ) B,n ( ˆX s ) ds = ds r r H B (y) B,n (y) dπ s (x, ) dµ which tends to, as n, by the dominated convergence theorem. We have found that (y)µ(dy), I n in L 1 (Ω, P).

24 The basic identity 24 Let us fix T >. Let X and Y are two mild solutions as in the uniqueness theorem. From the previous Itô-Tanaka formula we have, for t [, T τ X τy ], P-a.s., X t Y t = u (Y t ) u (X t ) + (λ A) e (t s)a (u (X s ) u (Y s )) ds + e (t s)a (Du (X s ) Du (Y s ))dw s. and after several computations we get that X = Y on [, T τ X τy ]. Remark. Why we say uniqueness for x H, µ-a.s.? A crucial point is to prove T H ( n 1 1 λ 1 δ n for some γ > 1. Here µ x s is the law of Z(s, x). D 2 u (n) (y) 2) 2γ µ x s (dy)ds <,

25 25 This is like T R s f (x)ds C x,t,p f L p (µ), (22) for any non-negative Borel function f (here R t is the OU Markov semigroup). We can prove this only for x H µ-a.s. On the other hand, in finite dimension this holds for any x R d, with p > c(d).

26 26 An application of our local uniqueness result: existence of strong solutions By combining our local uniqueness result and an infinite-dimensional version of the Yamada-Watanabe theorem (for such Yamada-Watanabe theorem, see Ondreját, M., Dissertationes Math. 24, and Liu, W., Röckner, M., Introduction to Stochastic Partial Differential Equations) we obtain Theorem Let us consider equation (1) and assume (H1), (H2) and (H3). Moreover, suppose that there exist C >, p >, such that B(y + z), y C ( y 2 + e p z + 1 ), y, z H. (23) Then, for µ-a.e. x H (where µ = (, 1 2 A 1 )), our equation (1) has a strong mild solution. Moreover, this solution is pathwise unique.

27 27 Sketch of Proof We only have to prove existence of strong solution for µ-a.e. x H. We will again consider truncated bounded drifts B = B 1 B(,), 1. By the main result in [DFPR12] there exists a Borel set G H with µ( G) = 1 such that for any x G we have pathwise uniqueness for each stochastic equation dx t = (AX t + B (X t ))dt + dw t, X = x H, 1. (24) Let x G. By the Girsanov theorem there exists (a unique in law) weak mild solution X = (X (t)) = (X (t)) t for each stochastic equation (24).

28 28 Therefore we can apply a generalization of the Yamada-Watanabe theorem to (24) when x G. Let us fix any filtered probability space (Ω, F, (F t ), P) on which there is defined a cylindrical (F t )-Wiener process W. By the Yamada-Watanabe theorem, for any 1, on the fixed filtered probability space above there exists a (unique) strong mild solution X to (24). Moreover, since B (x) = B +k (x), x B(, ), k 1, we have by our local uniqueness result that, P-a.s., τ := τ X and X (t τ ) = X +k (t τ ), t. = τx +k, k 1, 1,

29 29 It is enough to construct the strong solution X on [, T] for a fixed T >. We define an H-valued stochastic process X on Ω = 1 {τ > T} as X(t)(ω) := X (t)(ω), t [, T], if ω {τ > T} (we set X t (ω) = if ω Ω, t [, T]). Then X(t) is well defined. It is not difficult to prove that X is a strong mild solution on [, T] if we show that (this will imply that P(Ω ) = 1). lim P(τ > T) = 1 (25)