Kolmogorov equations for stochastic PDE s with multiplicative noise

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1 Kolmogorov equations for stochastic PDE s with multiplicative noise Giuseppe Da Prato 1 Scuola Normale Superiore, Pisa, Italy 1 Introduction We are here concerned with the following stochastic differential equation in the ilbert space = L 2 (, 1), dx(t, ξ) = Dξ 2 X(t, ξ)dt + g(x(t, ξ))dw (t, ξ), t, ξ [, 1, X(t, ) = X(t, 1) =, t, (1) X(, ξ) = x(ξ), x, ξ [, 1, where g is a real function of class C 2 bounded together with its derivatives of order less or equal to 2 and W is a cylindrical Wiener process in (see below for a precise definition). Existence and uniqueness of a solution of (1) are well known, see [16, [7. Let us denote by R t the corresponding transition semigroup, R t ϕ(x) = E[ϕ(X(t, x)), t, x, (2) where ϕ is a real bounded Borel function and X(t, x) is the solution of equation (1). The Kolmogorov equation corresponding to (1) reads as follows, u t (t, x) = 1 2 Tr [σ2 (x)u xx (t, x) + Ax, u x (t, x), t, x D(A), (3) u(, x) = ϕ(x), x, where A is the linear operator, Ax = D 2 ξ, x D(A) = 2 (, 1) 1 (, 1), and for any x L 2 (, 1) the symmetric Nemitskii operator σ(x) L(L 2 (, 1)) is defined by

2 2 Giuseppe Da Prato [σ(x)y(ξ) = g(x(ξ))y(ξ), y L 2 (, 1), ξ [, 1. (4) There is an increasing interest on infinite dimensional Kolmogorov equations, see the monographs [15, [5, [13, [8 (and references therein) and the papers [1,[17, [2, [3. In particular, in [13 the case when the noise is additive is mainly considered with the exception of chapters 6 and 7. More precisely, Chapter 6 is devoted to ölder continuous perturbations of the infinite dimensional eat semigroup, see also some recent developpements in this direction in [2 and [3. In chapter 7 of [13 the case when coefficients are of class C 3 is considered. Notice that in equation (1) the multiplicative noise can be written as σ(x)dw (t) where for any x L 2 (, 1) the operator σ(x) L(L 2 (, 1)) is operator defined by (4). Thus, in spite of the fact that g is C 2, the mapping σ(x) is only once Gateaux differentiable (except when g is constant). So, the method in [13, Chapter 7 does not work and some new technique has to be used. The first main result of the paper is that if ϕ Cb 3 () there is a smooth solution of (3), see Theorem 13. It is well known that a candidate for the solution of (3) is given by u(t, x) := R t ϕ(x) = E[ϕ(X(t, x)), (5) where R t is the transition semigroup defined by (2). So, the proof of existence of a solution of (3) will consists in showing (by justifying the chain rule) that u(t, x) is twice differentiable, that the trace of σ 2 D 2 u(t, x) is finite and that equation (3) is fulfilled. owever, these computations are not straightforward since σ is neither regular nor of trace class. This idea was applied in the case of reaction diffusion equations with additive noise, see [6 and in the completely different situation of Navier Stokes equations with additive noise, see [9. We notice that the regularity we get for the solution of (3) is not enough to apply the Itô formula and so, to prove the uniqueness of a smoth solution of (3). This would require an additional job which we plan to make in a future paper. In the second part of the paper we assume that 1/g is bounded so that there is a unique invariant measure µ, see [16. We study here the Kolmogorov equation (2) in the space L 2 (, µ). It is well known that the transition semigroup R t can be uniquely extended to a strongly continuous semigroup of contractions in L 2 (, µ). We shall still denote by R t this extension and by L µ the infinitesimal generator of R t. As a second main result of the paper we construct a core Γ for the generator L µ consisting of regular functions and show that on Γ the operator L µ is in fact a differential operator given by L µ ϕ(x) = 1 2 Tr [σ2 (x)d 2 ϕ(x) + Ax, Dϕ(x), x D(A), ϕ Γ.

3 Kolmogorov equations for stochastic PDE s with multiplicative noise 3 As it was pointed out in several previous situations, see e.g. [8 and references therein, to have an explicit expression of L µ on the core Γ allows to prove easily the so called identité du carré des champs, L µ ϕ ϕ dµ = 1 σ(x)dϕ 2 dµ, ϕ D(L µ ). (6) 2 Using (6) it is possible to prove that the derivative operator is closable in L 2 (, µ). This allows to define the Sobolev space W 1,2 (, µ) and to show that the domain of L µ is included in W 1,2 (, µ). Finally, we prove the Poincaré inequality, by generalizing a result proved in [1 in the case of additive noise. As a standard consequence we obtain the spectral gap of L µ and the exponential convergence to equilibrium of R t. It would be interesting to consider the more general problem, dx(t, ξ) = (Dξ 2 X(t, ξ) + f(x(t, ξ)))dt + g(x(t, ξ))dw (t, ξ), ξ [, 1, X(t, ) = X(t, 1) =, t X(, ξ) = x(ξ), x, ξ [, 1, (7) where f is a suitable real function. owever, problem (7) does not seem to be a straightforward generalization of (1). It will be the object of a future research. 1.1 Notations We denote by the ilbert space L 2 (, 1) (norm, inner product, ). When there is the danger of confusion between the norm and the absolute value of a function x we shall write x L 2 (,1) instead of x. Moreover L() (norm ) will represent the Banach algebra of all linear bounded operators in and L 1 () (norm L1()) the space of all trace class operators in. We recall that T = sup{ T x : x, x = 1}, T L(). For any ilbert space K (norm, inner product, ), we denote by C b (; K) the linear space of all continuous and bounded mappings ϕ: K. C b (; K) endowed with the norm ϕ = sup ϕ(x), x K ϕ C b (; K), is a Banach space. Moreover Cb 1(; K) will represent the subspace of C b(; K) of all functions ϕ: K which are Fréchet differentiable on with a continuous and bounded derivative Dϕ. The space Cb k (; K) for k 2 are defined analogously. We shall write Cb i(; R) = Ci b (), i N.

4 4 Giuseppe Da Prato If ϕ Cb 1 () and x, we shall identify Dϕ(x) with the unique element h of such that Dϕ(x)y = h, y, x, y. If ϕ C 2 b () and x, we shall identify D2 ϕ(x) with the unique linear operator T L() such that Dϕ(x)(y, z) = T y, z, x, y, z. 1.2 An extension of Gronwall s lemma The following result is a generalization of a well known result. Lemma 1. Assume that f : [, + ) [, + ) fulfills the inequality, f(t) a(t) + b t (t s) 1/2 f(s)ds, t, (8) where a is continuous nonnegative and b is a nonnegative constant. Then we have, f(t) a(t) + b + t t (t s) 1/2 a(s)ds e (t s)πb2 [a(s) + b If, in particular, a(t) = a we have and Proof. We write (8) as f(t) ae πb2t + 2ab t f(t) 3ae πb2 t s (s σ) 1/2 a(σ)dσ. (9) s 1/2 e πb2 (t s) ds t. (1) t. (11) f a + bψ 1/2 f, (12) where ψ 1/2 (t) = t 1/2 and denotes the convolution ( 1 ). Taking the convolution of both sides of (1) with ψ 1/2 and taking into account that yields (ψ 1/2 ψ 1/2 )(t) = Substituting this in (12) yields 1 (f g)(t) = R t f(t s)g(s)ds. t (t s) 1/2 s 1/2 ds = π, ψ 1/2 f a ψ 1/2 + πb(1 f). (13)

5 Kolmogorov equations for stochastic PDE s with multiplicative noise 5 which is equivalent to Consequently f(t) a(t) + b f(t) a(t) + b t f a + b(a ψ 1/2 ) + πb 2 (1 f), (14) t (t s) 1/2 a(s)ds + t (t s) 1/2 a(s)ds + πb 2 f(s)ds. (15) t e (t s)πb2 [a(s) + b s (s σ) 1/2 a(σ)dσ Now (9) follows from the classical Gronwall lemma. Finally, (1) is clear and by (1) we have ( f(t) ae πb2 t 1 + 2b which yields (11). t ) s 1/2 e πb2s ds ae πb2 t ( 1 + 2b ) s 1/2 e πb2s ds, 2 Existence and uniqueness of solutions 2.1 The abstract setting Let us write problem (1) in an abstract form introducing the linear self adjoint operator A: D(A), { Ax = D 2 ξ x, x D(A), D(A) = 2 (, 1) 1 (, 1), where i (, 1), i = 1, 2 denote the usual Sobolev spaces and 1 (, 1) = {x 1 (, 1) : x() = x(1) = }. We define moreover the (generally) nonlinear operator σ : L() by setting, [σ(x)y(ξ) = g(x(ξ))y(ξ), ξ [, 1, x, y. We denote by (e k ) the complete orthonormal sistem in consisting of the eigenfunctions of A, 2 e k (ξ) = sin kπξ, ξ [, 1, k N, π

6 6 Giuseppe Da Prato so that Notice that Ae k = k 2 π 2 e k, k N. e ta e π2t, t. Finally, we introduce the cylindrical white noise, W (t) = e k β k (t), t, (16) where (β k ) is a sequence of mutually independent standard Brownian motions on a filtered probability space (Ω, F, (F t ) t, P). Now we can write problem (1) as follows, dx = AXdt + σ(x) dw (t) = AXdt + [g(x)e k dβ k (t), (17) X() = x. We shall solve equation (17) in the space C W ([, T, ) of all mean square continuous adapted (to the filtration (F) t ) stochastic process X( ) defined in [, T and taking values in. It is well known that C W ([, T, ), endowed with the norm ( is a Banach space. X CW ([,T,) = 1/2 sup E( X(t) )) 2, t [,T Definition 2. A mild solution of equation (17) is a process X C W ([, T, ) such that X(t) = e ta x + t e (t s)a σ(x(s))dw (s), t, x. (18) In the following we shall denote by X(, x) the solution of (18). An important rôle will be played by the stochastic convolution, W X (t) = t e (t s)a σ(x(s))dw (s) = t e (t s)a [g(x(s))e k dβ k (s), where X C W ([, T, ). As we shall see, though the the cylindrical white noise (16) does not leave in (see e.g. [11, 4.3.1), the stochastic convolution W X (t) does. In order to study basic properties of W X (t) it is useful to introduce the function

7 Kolmogorov equations for stochastic PDE s with multiplicative noise 7 Notice that F (t) e t + = e t (1 + 1 e ty2 F (t) = e th2, t >. (19) h=1 e tx2 dx = e t (1 + ) y y2 + 1 dy e (1 t + 1 ) e t(x2 1) dx ) e ty2 dy. So, F (t) e t ( 1 + 2t 1/2) 4t 1/2 e t/2, t >. (2) Lemma 3. Let X C W ([, T, ). Then we have E( W X (t) 2 ) = t e 2π2 h 2 (t s) E( g(x(s))e h 2 )ds. (21) h=1 Proof. Taking into account the independence of the (β k ) we have, E( W X (t) 2 ) = t E( e (t s)a [g(x(s))e k 2 )ds. Using the Parseval identity we find, e (t s)a [g(x(s))e k 2 = e (t s)a [g(x(s))e k, e h 2 h, = e 2(t s)π2 h 2 g(x(s))e k, e h 2 = e 2(t s)π2 h 2 e k, g(x(s))e h 2 h, h, = e 2(t s)π2 h 2 σ(x(s))e h 2. h=1 So, (21) follows. Proposition 4. Let X C W ([, T, ). Then we have E( W X (t) 2 ) 8 π g 2, t. (22) Moreover, for all X, Y C W ([, T, ) we have, E( W X (t) W Y (t) 2 ) = 8 g 2 1 π 2π t e (t s)π2 (t s) 1/2 E[ X(s) Y (s) 2 )ds. (23)

8 8 Giuseppe Da Prato Proof. By (21) we have, recalling that e k (ξ) 2 2 π (2), E W X (t) 2 2 g 2 π t F (2π 2 s)ds 8 g 2 π and (22) follows. The proof of (23) is similar. and taking into account e π2s s 1/2 ds 2.2 Existence and uniqueness The following result is well known, see e.g. [16, we present however the short proof for the reader s convenience. Proposition 5. For any x there exists a unique solution X(, x) of equation (18). Proof. Write equation (18) in the form X = e ta x + Λ(X), X C W ([, T, ), where Λ(X)(t) = W X (t), t [, T. Then by (22) it follows that Λ maps C W ([, T, ) in itself. Let moreover X, Y C W ([, T, ). Then by (23) it follows that, Λ(X)(t) Λ(Y )(t) 8 g 2 1 π π t e (t s)π2 (t s) 1/2 ds X Y CW ([,T,) 16 g 2 1 π 2π t1/2 X Y CW ([,T,). Now let T 1 (, T be such that 16 g 2 1 π 2π T 1/2 1 < 1. Then Γ is a contraction on C W ([, T 1, ). Therefore, equation (18) has a unique solution on [, T 1. By a similar argument, one can show existence and uniqueness on [T 1, 2T 1 and so on. 2.3 Galerkin approximations It is useful to consider Galerkin approximations of equation (18). For any n N we denote by P n the projector n P n x = x, e k e k, x and set A n = AP n. Then we consider the equation

9 Kolmogorov equations for stochastic PDE s with multiplicative noise 9 X n (t, x) = e tan x + n The following result is standard. t e (t s)an [g(x n (s, x))e k dβ k (s). (24) Proposition 6. For any T >, x and n N, there exists a unique solution X n (, x) of equation (24). Moreover, lim n Xn (, x) = X(, x), in C W ([, T, ), (25) where X(, x) is the solution of (18). 3 Kolmogorov equation 3.1 Setting of the problem We are here concerned with the following Kolmogorov equation, u t (t, x) = 1 2 Tr [σ2 (x)u xx (t, x) + Ax, u x (t, x), t, x D(A), u(, x) = ϕ(x), x. (26) We are going to show that when the initial datum ϕ is sufficiently regular equation (26) has a solution in a classical sense. As it is well known, a candidate for the solution u(t, x) of (26) is provided by the formula u(t, x) = E[ϕ(X(t, x)), ϕ C b (), t, x, (27) where X(t, x) is the solution of (18). We shall check that, under suitable assumptions on ϕ, formula (27) produces in fact a solution of (26). We shall need to consider the approximating equation u n t (t, x) = 1 2 Tr [P nσ 2 (x)u n xx(t, x) + A n x, u n x(t, x) (28) u n (, x) = ϕ(p n x), which has a unique strict solution which we denote by u n (t, x). 3.2 Estimates for derivatives of X(t, x) This subsection is devoted to establish some estimates concerning the derivatives X x and X xx, which will be used later. We start from the directional derivative η z 1 (t, x) := X x (t, x)z = lim (X(t, x + ɛz) X(t, x)) ɛ ɛ

10 1 Giuseppe Da Prato where z. By using Galerkin approximations it is not difficult to show that the diretional derivative η z (t, x) does exist and it is the solution of the equation, t η z (t, x) = e ta z + e (t s)a [g (X(s, x))η z (s, x)e k dβ k (s). (29) Lemma 7. There exists two positive constants a 1 and λ 1 such that Proof. We have E( η z (t, x) 2 ) = e ta z 2 + E( η z (t, x) 2 ) a 1 z 2 e λ1t t, x. (3) E t e (t s)a [g (Y (s, x))η z (s, x)e k 2 ds. Arguing as in the proof of Lemma 3 and taking into account (21), we see that, E( η z (t, x) 2 ) e ta z π g 2 t F (2π 2 (t s))e( η z (s, x) 2 )ds e ta z g 2 2π 2 By Lemma 1 it follows that t (t s) 1/2 e (t s)/2 E( η z (s, x) 2 )ds. E( η z (t, x) 2 ) e ta z 2 e 16 g 2 2π 2 t 1/2, t and the conclusion follows. We want now to estimate ζ z (t, x) := X xx (t, x)(z, z) where 1 X xx (t, x)(z, z) = lim ɛ ɛ (ηz (t, x + ɛz) η z (t, x)) and z. Formally ζ z (t, x) is the solution of the equation, t ζ z (t, x) = e (t s)a [g (Y (s, x))ζ z (s, x)e k dβ k (s) + t ere a problem arises since the term e (t s)a [g (Y (s, x))(η z (s, x)) 2 e k dβ k (s). (31) g (Y (s, x))(η z (s, x)) 2 e k, (32) which appears in the second integral, belongs to L 4 (, 1) and not to L 2 (, 1) in general. For this reason we first need an estimate for E( η z (t, x) 4 L 4 (,1) ) = E( [ηz (t, x) 2 2 L 2 (,1) ). To get this estimate we shall proceed in two steps.

11 Kolmogorov equations for stochastic PDE s with multiplicative noise 11 (i) We shall estimate E( η z (t, x) 4 L 2 (,1) ). (ii) We shall estimate E( ( A) 1/8 η z (t, x) 4 L 2 (,1) ). Then we notice that, by the Sobolev embedding theorem, we have D(( A) 1/8 ) L 4 (, 1) and so we end up with the required estimate for E( η z (t, x) 4 L 4 (,1) ). Let us write (29) as where η z (t, x) = e ta z + Φ(t s) = e (t s)a σ(s) = t Φ(t s)dw (s), (33) e (t s)a [g (X(s, x))η z (s, x)e k. (34) We shall use the following Burkholder estimate, see [11. [ t 4 [ ( t ) 2 E Φ(t s)dw (s) ce Φ(t s) 2 Sds, (35) where c is a given positive constant and Φ(t s) 2 S = = h, e (t s)a [g (X(s, x))η z (s, x)e k, e h 2 e 2π2 h 2 (t s) g (X(s, x))η z (s, x)e k, e h 2 h, = e 2π2 h 2 (t s) g (X(s, x))η z (s, x)e h 2 h=1 (36) 2 π g 2 1F (2π 2 (t s)) η z (s, x) 2 32 π 2 g 2 1(t s) 1/2 η z (s, x) 2. where we have used (21). Now we are ready to prove Lemma 8. Let z L 2 (, 1). Then there exists constants a 2 > and λ 2 > such that E( η z (t, x) 4 L 2 (,1) ) a 2e λ2t z 4 L 2 (,1), t, x. (37)

12 12 Giuseppe Da Prato Proof. Let z L 2 (, 1). By (33) we have [ t E( η z (t, x) 4 L 2 (,1) ) 8 eta z 4 L 2 (,1) + 8E Φ(t s)dw (s) which, taking into account (35), yields E( η z (t, x) 4 L 2 (,1) ) 8 eta z 4 L 2 (,1) + 8cE [ ( t Let us estimate the term J := E J 21 π 4 = 21 π 4 g 4 1 [ ( t 4 L 2 (,1) ) 2 Φ(t s) 2 Sds. (38) ) 2 Φ(t s) 2 Sds. We have by (36) [ ( t ) 2 g 4 1E (t s) 1/2 η z (s, x) 2 L 2 (,1) ds t t (t s) 1/2 (t s 1 ) 1/2 [ E η z (s, x) 2 L 2 (,1) ηz (s 1, x) 2 L 2 (,1) dsds 1 29 π 4 g π 4 g 4 1 t t t t 211 π 4 g 4 1t 1/2 t Substituting in (38) yields (t s) 1/2 (t s 1 ) 1/2 E (t s) 1/2 (t s 1 ) 1/2 E (t s) 1/2 E [ η z (s, x) 4 L 2 (,1) dsds 1 [ η z (s 1, x) 4 L 2 (,1) dsds 1 [ η z (s, x) 4 L 2 (,1) ds.. E( η z (t, x) 4 L 2 (,1) ) 8 eta z 4 L 2 (,1) +c 214 π 4 g 4 1t 1/2 t (t s) 1/2 E [ η z (s, x) 4 L 2 (,1) ds. Now by the Gronwall Lemma it follows that there exist positive constants ρ, l such that t E( η z (t, x) 4 L 2 (,1) ) 8 eta z 4 L 2 (,1) + ρ (t s) 1/2 e l(t s) e sa z 4 L 2 (,1) ds, which implies the conclusion. (39)

13 Kolmogorov equations for stochastic PDE s with multiplicative noise 13 Lemma 9. Let z L 2 (, 1). Then there exists constants a 3 > and λ 3 > such that E( ( A) 1/8 η z (t, x) 4 L 2 (,1) ) a 3e λ3t ( A) 1/8 e ta z 4 L 2 (,1), t, x. (4) Proof. Let z L 2 (, 1). Then we have where E( ( A) 1/8 η z (t, x) 4 L 2 (,1) ) 8 ( A)1/8 e ta z 4 L 2 (,1) +8E [ t Φ 1 (t s)dw (s) Φ 1 (t s) = ( A) 1/8 e (t s)a σ(s) = We have Φ 1 (t s) 2 S = = = h, 4 L 2 (,1). ( A) 1/8 e (t s)a [g (X(s, x))η z (s, x)e k. ( A) 1/8 e (t s)a [g (X(s, x))η z (s, x)e k, e h 2 (πh) 1/2 e 2π2 h 2 (t s) g (X(s, x))η z (s, x)e k, e h 2 h, (πh) 1/2 e 2π2 h 2 (t s) g (X(s, x))η z (s, x)e h 2. h=1 It is not difficult to show that there is a positive constant d 1 such that Now we have Φ 1 (t s) 2 S d 1 (t s) 3/4 η z (s, x) 2. (41) E( ( A) 1/8 η z (t, x) 4 L 2 (,1) ) 8 ( A)1/8 e ta z 4 L 2 (,1) +8cE [ ( t ) 2 Φ 1 (t s) 2 Sds. (42) By proceeding as in the proof of the previous lemma we find that there is d 2 > such that [ ( t ) 2 E Φ 1 (t s) 2 Sds t [ d 2 g 4 1t 1/4 (t s) 3/4 E η z (s, x) 4 L 2 (,1) ds

14 14 Giuseppe Da Prato which yields E( ( A) 1/8 η z (t, x) 4 L 2 (,1) ) 8 ( A)1/8 e ta z 4 L 2 (,1) t [ +d 2 t 1/2 (t s) 3/4 E η z (s, x) 4 L 2 (,1) ds. So, the conclusion follows from Lemma 8. Now, using the Sobolev embedding D(( A) 1/8 ) L 4 (, 1) we can conclude that Lemma 1. Let z L 2 (, 1). Then there exists constants a 4 > and λ 4 > such that E( η z (t, x) 4 L 4 (,1) ) a 4e λ4t z 4 L 2 (,1), t, x. (43) Now we are in position to estimate ζ z for z L 2 (, 1). Lemma 11. There exists constants a 5 > and λ 5 > such that for all z L 2 (, 1), we have E( ζ z (t, x) 2 ) a 5 z 4 L 2 (,1)) eλ5t, t, x. (44) Proof. Let z L 2 (, 1). Using Galerkin approximations we can show that (31) holds. From (31) we deduce that, E( ζ z (t, x) 2 ) 2E +2E t t = 2J 1 + 2J 2. Arguing as in the proof of Lemma 3 we find J 1 = E h, e (t s)a [g (X(s, x))ζ z (s, x)e k 2 ds e (t s)a [g (X(s, x))(η z (s, x)) 2 e k 2 ds: e (t s)a [g (X(s, x))ζ z (s, x)e k, e h 2 = E e 2(t s)π2 h 2 g (X(s, x))ζ z (s, x)e k, e h 2 h, = E e 2(t s)π2 h 2 g (X(s, x))ζ z (s, x)e h 2 h=1 (45) 2 π g 2 1F (2(t s)π 2 )E ζ z (s, x) 2 4 π 2 (t s) 1/2 E ζ z (s, x) 2.

15 Kolmogorov equations for stochastic PDE s with multiplicative noise 15 Similarly for J 2 we find, taking into account Lemma 1, J 2 = E e (t s)a [g (X(s, x))(η z (s, x)) 2 e k, e h 2 h, = E e 2(t s)π2 h 2 g (X(s, x))(η z (s, x)) 2 e k, e h 2 h, = E e 2(t s)π2 h 2 g (X(s, x))(η z (s, x)) 2 e h 2 h=1 (46) 2 π g 2 1F (2(t s)π 2 )E (η z (s, x)) π 2 (t s) 1/2 E (η z (s, x)) 2 2 a 4 4 π 2 (t s) 1/2 e λ4s z 4 L 2 (,1). Now the conclusion follows from (45), (46) and the Gronwall lemma. Remark 12. By Lemma 11 it follows that if ϕ Cb 2 () the function u(t, ) possesses bounded second order derivatives in all directions of for any t. So, it is Frécher differentiable and belongs to Cb 1 () (more precisely to C 1+ε b () for all ε (, 1)). 3.3 Strict solutions of the Kolmogorov equation We are now in position to show existence of a strict solution u(t, x) (in the sense that u(t, x) fulfills conditions (i)-(iv) of Proposition 13 below) of equation (26) for all ϕ Cb 2 (). Let us define, λ = 1 2 max{λ i : i = 1,.., 5} and κ = 1 2 max{a1/2 i : i = 1,.., 5} Theorem 13. Assume that ϕ C 2 b (), D2 ϕ(x) is of trace class for any x and Tr [D 2 ϕ C b (). Let u(t, x) = E[ϕ(X(t, x)), t, x, where X(t, x) is the mild solution of (17). Then the following statements hold. (i) For all t, u(t, ) C 1 b () and possesses second order derivatives in all directions of.

16 16 Giuseppe Da Prato (ii) For all t > and any x we have and u x (t, x) κ e λt ϕ 1 (47) u xx (t, x) κ e λt ϕ 2. (48) (iii) There exists κ 1 > such that for all t and any x we have Tr [σ 2 (x)u xx (t, x) = u xx (t, x)(σ(x)e k, σ(x)e k ) κ 1 e λt ϕ 2 (1 + sup D 2 ϕ(x) L1()). x (iv) For all x D(A), u(, x) is differentiable in (, + ) and fulfills (26). Proof. Let us prove (i). For any x, z and t we have Therefore By Lemma 7 it follows that u x (t, x), z = E[ Dϕ(X(t, x), η z (t, x) u x (t, x), z ϕ 1 η z (t, x). u x (t, x), z ϕ 1 a 1/2 1 e λ1t/2. (49) Thus (47) follows from the arbitrariness of z. Moreover u(t, ) Cb 1 () in view of Remark 12. Let us prove (ii). For any x, z and t we have u xx (t, x)z, z = E[ Dϕ(X(t, x), ζ z (t, x) + E[ D 2 ϕ(x(t, x)η z (t, x), η z (t, x). Therefore u xx (t, x)z, z ϕ 1 E[ ζ z (t, x) + ϕ 2 E[ η z (t, x) 2 ϕ 1 a5 e λ5t/2 z 2 + ϕ 2 a 1 e λ1t/2 z 2 and (ii) follows. Let us prove (iii). For any x, z and t we have Tr [σ 2 (x)u xx (t, x) = = u xx (t, x)(σ(x)e k, σ(x)e k ) E[ Dϕ(X(t, x), ζ σ(x)e k (t, x) + Tr E[σ 2 (x)x x (t, x)d 2 ϕ(x(t, x)x x(t, x) := J 1 (t, x) + J 2 (t, x).

17 Kolmogorov equations for stochastic PDE s with multiplicative noise 17 Therefore J 2 (t, x) a 1 e λ1t D 2 ϕ S. Concerning J 1 we have J 1 (t, x) ϕ 1 T (t, x) where T (t, x) = ζ σ(x)e k (t, x). Then one can checks that T (t, x) is the solution of the equation T (t, x) = t t + e (t s)a [g (X(s, x))t (s, x)dw (s) e (t s)a [g (X(s, x))k(s, x)dw (s), (5) where K(t, x) = (η σ(x)e k (t, x)) 2. Using estimate (39), it is not difficult to show that equation (5) has a solution and estimate (49) holds. Let us prove finally (iv). Assume that x D(A) and let u n (t, x) be the solution of (28). Then, taking into account that estimates from Lemmas 7 and 11 can also be proved for the function u n with constants independent of n, it is not difficult to check that, lim n un (t, x) = u(t, x), t >, x, and lim n un x(t, x) = u x (t, x), t >, x, lim Tr n [σ(x)cσ(x)un xx(t, x) = Tr [σ(x)cσ(x)u xx (t, x), t >, x. Consequently, lim n un t (t, x) = u t (t, x), t >, x D(A), and the conclusion follows. We consider finally the elliptic Kolmogorov equation λϕ(x) 1 2 Tr [σ2 (x)ϕ xx (x) Ax, ϕ x (x) = f(x), x D(A), (51) where λ > and f C b () are given.

18 18 Giuseppe Da Prato Theorem 14. Assume that λ > λ, f C 2 b (), D2 f(x) is of trace class for any x and Tr [D 2 f C b (). Define ϕ(x) = Then the following statements hold. e λt E[f(X(t, x))dt, t, x, (i) ϕ Cb 1 () and possesses second order derivatives in all directions of. (ii) For all x we have and u x (t, x) κ λ λ ϕ 1 (52) u xx (t, x) κ λ λ ϕ 2. (53) (iii) There exists κ 1 > such that for all x we have Tr [σ 2 (x)u xx (t, x) = u xx (t, x)(σ(x)e k, σ(x)e k ) (ii) We have ϕ x (x) κ 1 λ λ ϕ 2(1 + D 2 ϕ S ). a1 (54) λ λ f 1, x, (55) and Tr [σ 2 (x)ϕ xx (t, x) κ (λ l) f 2, x. (56) (iv) For all x D(A) the equation (51) is fulfilled. Proof. The conclusion follows from Proposition 13 and estimates (47), (48) and (49). 3.4 The Kolmogorov operator It is well known that the semigroup R t is not in general strongly continuous in C b (). owever, we can define its infinitesimal generator by proceeding as in [4. Namely, for any λ > and any f C b () we define F λ (f)(x) = e λt R t f(x)dt, x.

19 Kolmogorov equations for stochastic PDE s with multiplicative noise 19 Proposition 15. For any f C b () and any λ > we have F λ (f) C b () and the following estimate holds F λ (f) 1 λ f. (57) Moreover there exists a unique closed operator L: D(L) C b () C b () such that for any λ > and any f C b () we have F λ (f) = R(λ, L)f. Proof. Let first f Cb 1(); then it is obvious that if F λ(f) C b () the inequality (57) holds. Moreover for all x, y we have F λ (f)(x) F λ (f)(y) On the other hand we have X(t, x) X(t, y) = e λt E( f(x(t, x)) f(x(t, y)) )dt f 1 e λt E X(t, x) X(t, y) dt. 1 so that, recalling Lemma 7, we find X x (t, (1 r)x + ry)(x y)dr, (58) X(t, x) X(t, y) a 1 e 1 2 λ1t x y. (59) Now, substituting this inequality in (58) yields F λ (f)(x) F λ (f)(y) a 1 e ( 1 2 λ1 λ)t dt x y. Thus, if λ > 1 2 λ 1 we have proved that F λ (f) C b () (it is even Lipschitz). Since C 1 b () is dense in C b() we can conclude that F λ (f) C b () for all f C b () (and λ > 1 2 λ 1). Now it is easy to see that F λ fulfills the resolvent identity F λ F µ = (µ λ)f λ F µ, λ, µ >. So, by a classical result, see e. g. [18, there exists a unique closed operator L: D(L) C b () C b () such that for any λ > 1 2 λ 1 and any f C b () we have F λ (f) = R(λ, L)f. Finally, by (57) we see that L is m-dissipative so that condition λ > 1 2 λ 1 can be replaced by λ >. Remark 16. Assume that f C 2 b (), D2 f(x) is of trace class for any x and Tr [D 2 f C b (). Let moreover λ > and ϕ = R(λ, L). Then by Proposition 14 it follows that Lϕ(x) = 1 2 Tr [σ2 (x)ϕ xx (x) + Ax, ϕ x (x), x D(A). (6)

20 2 Giuseppe Da Prato Now we are going to prove, following an argument in [1, that if ϕ R(λ, L)(C 2 b ()) we have ϕ2 D(L). Proposition 17. Assume that f C 2 b (), D2 f(x) is of trace class for any x and Tr [D 2 f C b (). Let moreover λ > and ϕ = R(λ, L). Then ϕ 2 D(L) and L(ϕ 2 ) = 2ϕ Lϕ + σdϕ 2. (61) Proof. Let L n be the approximating Kolmogorov operator L n ϕ(x) = 1 2 Tr [σ2 (x)p n ϕ xx (x) + A n x, ϕ x (x), ϕ C b (), x (62) and let ϕ n = R(λ, L n )f. Then, by a straightforward computation, it follows that L n ((ϕ n ) 2 ) = 2ϕ n L n ϕ n + σp n Dϕ n 2. Now, multiplying both sides of the equation λϕ n L n ϕ n = f by ϕ n, yields which is equivalent to Therefore, Letting n yields λ(ϕ n ) 2 L n ϕ n ϕ n = fϕ n, 2λ(ϕ n ) 2 L n ((ϕ n ) 2 ) = 2fϕ n σp n Dϕ n 2. (ϕ n ) 2 = R(2λ, L n )(2fϕ n σp n Dϕ n 2 ). ϕ 2 = R(2λ, L)(2fϕ σdϕ 2 ). Consequently which yields (61). 2λϕ 2 Lϕ 2 = 2fϕ σ(x)dϕ 2, 4 Invariant measures 4.1 Existence and uniqueness We denote by P() the set of all Borel probability measures on. We recall that a probability measure µ P() is said to be invariant for the transition semigroup R t defined by (7) if R t ϕdµ = ϕdµ for all ϕ C b (). (63)

21 Kolmogorov equations for stochastic PDE s with multiplicative noise 21 Theorem 18. There is an invariant measure µ for R t. Moreover, for any β [, 1/4) we have ( A) β x 2 µ(dx) < +. (64) Finally, if 1/g is bounded the invariant measure µ is unique. Proof. Let X(t, x) be the solution of (18). Using Lemma 3 and inequality (21), we find that E( X(t, x) 2 ) 2e 2π2t x h=1 2e 2π2t x 2 + g 2 t t e 2π2 h 2 (t s) E( σ(x(s, x))e h 2 )ds F (2π 2 (t s))ds So, 2e 2π2t x g 2 t e π2 (t s) (2π 2 (t s)) 1/2 ds. E( X(t, x) 2 ) 2e 2π2t x π g 2. (65) Now let β (, 1/4). Using the well known estimate where c β is a suitable constants, we find, E( ( A) β X(t, x) 2 ) t 2c β t 2β e 2π2t x c β t 2β e 2π2t x g 2 where F β is defined by ( A) β e ta c β t β e π2t, t, (66) F β (t) = (πh) 4β e 2π2 h 2 (t s) E( σ(x(s, x))e h 2 )ds h=1 t F β (2π 2 (t s))ds h 4β e h2t, t >. h=1 It is not difficult to show that there is k β > such that Now we have F β (t) k α t 1/2 2β e t, t.

22 22 Giuseppe Da Prato E( ( A) β X(t, x) 2 ) 2c β t 2β e 2π2t x 2 t +2 g 2 k β (2π 2 s) 1/2 2β e 2π2s ds (67) 2c β t 2β e 2π2t x 2 + c β k β g 2. Since the embedding D(A) is compact, the existence of an invariant measure follows from the Krylov Bogoliubov theorem. Let us show (64). Let γ > and set ϕ γ (x) = x γ x 2, x. Then ϕ γ C b () and, proceeding as in the proof of (65) we see that there exists a constant κ > (independent on λ) such that, R t (ϕ γ )(x) = E[ϕ γ (X(t, x)) e 2π2t ϕ γ (x) + κ. (68) Integrating both sides of (68) with respect to x over and taking into account the invariance of µ yields, ϕ γ (x)µ(dx) e 2π2 t ϕ γ (x)µ(dx) + κ. Therefore, there exists κ 1 > (independent on λ) such that ϕ γ (x)µ(dx) κ 1. Letting γ tend to yields, x 2 µ(dx) κ 1. (69) Now, integrating both sides of (67) with respect to x over and taking into account again the invariance of µ yields, ( A) β x 2 µ(dx) 2c β t 2β e 2π2t κ 1 + c β g 2, and so, (64) follows. Finally, if 1/g is bounded the uniqueness of µ follows from from the Doob Theorem since R t is irreducible and strong Feller by [16. Remark 19. More general results of existence of invariant measures can be found in the paper [7.

23 Kolmogorov equations for stochastic PDE s with multiplicative noise Existence of a core of smooth functions for L µ Let us fix an invariant measure µ for R t. It is well known that R t can be uniquely extended to a strongly continuous semigroup of contractions on L 2 (, µ) which we shall still denote by R t. The infinitesimal generator of R t in L 2 (, µ) will be denoted by L µ. Since R t is a contraction semigroup, L µ is m-dissipative in L 2 (, µ). In this subsection we want to define a core of L µ consisting of regular functions. Proposition 2. Set Λ = {ϕ C 2 b () : D 2 ϕ(x) L 1 () for all x and Tr [D 2 ϕ C b ()} and Γ := λ> R(λ, L)(Λ). Then Γ is a core for L µ. Moreover if ϕ Γ we have ϕ 2 D(L µ ) and the following identity holds L µ (ϕ 2 ) = 2ϕ L µ ϕ + σdϕ 2. (7) Proof. Let λ >. It is clear that any ϕ D(L) belongs to D(L µ ) as well, so that we have (λ L µ )(Γ ) = (λ L)(Γ ) Λ. Since Λ is dense in L 2 (, µ) (by a standard argument of monotone classes), we can conclude that (λ L µ )(Γ ) is dense in L 2 (, µ). Now the Lumer-Phillips theorem implies that Γ is a core for L µ. Finally, the last statement follows from (61). 5 The basic integration by parts formula In this section we assume that g 1 is bounded. We recall that in this case µ is the unique invariant measure of the transition semigroup R t. Proposition 21. The operator D : Γ L 2 (, µ) L 2 (, µ; ), ϕ Dϕ, (71) is uniquely extendible to a linear bounded operator D : D(L µ ) L 2 (, µ; ), where D(L µ ) is endowed with the graph norm of L µ. Moreover, the following identity holds L µ ϕ ϕ dµ = 1 σdϕ 2 dµ, ϕ D(L µ ). (72) 2

24 24 Giuseppe Da Prato Identity (72) is called in French identité du carré du champs. It will play an important rôle in what follows. Proof. Let ϕ Γ, then ϕ 2 D(L µ ) in view of Proposition 2 we have L µ ϕ 2 = 2ϕL µ ϕ + gdϕ 2. Integrating this identity with respect to µ over and taking into account that L µ ϕ 2 dµ = by the invariance of µ, implies(72) when ϕ Γ. Let now ϕ D(L µ ). Since Γ is a core for L µ, there exists a sequence {ϕ n } Γ such that ϕ n ϕ, L ϕ n L µ ϕ in L 2 (, µ). By (72) it follows that σd(ϕ n ϕ m ) 2 dµ = 2 L µ (ϕ n ϕ m ) (ϕ n ϕ m ) dµ. So, the sequence {σdϕ n } is Cauchy in L 2 (, µ; ) and the conclusion follows. Proposition 22. Let ϕ L 2 (, µ) and t. Then, for any T >, the linear operator σdr t : D(L µ ) L 2 (, µ) L 2 (, T ; L 2 (, µ; )), ϕ σdr t ϕ, is uniquely extendible to a linear bounded operator, still denoted by σdr t, from L 2 (, µ) into L 2 (, T ; L 2 (, µ; )). Moreover the following identity holds (R t ϕ) 2 dµ + t ds σdr s ϕ 2 dµ = ϕ 2 dµ. (73) Proof. We first establish (73) for ϕ D(L µ ). In this case we have d dt R tϕ = L µ ϕ. Multiplying scalarly this identity by R t ϕ, integrating with respect to µ over and using (73), yields, d (R t ϕ) 2 dµ + σdr s ϕ 2 dµ =. (74) dt Now (73) follows integrating (74) with respect to t. The case when ϕ L 2 (, µ) can be handled by approximating ϕ by elements of D(L µ ).

25 Kolmogorov equations for stochastic PDE s with multiplicative noise The Sobolev space W 1,2 (, µ) To define the Sobolev space we first show that the mapping D µ : Γ L 2 (, µ) L 2 (, µ; ), ϕ D µ ϕ (75) is closable. Notice the difference between the map D defined by (71) and the map D µ. The first one is a bounded operator in D(L µ ) (endowed with the graph norm of L µ ) whereas the second will be a closable operator in L 2 (, µ). To prove closability of D µ we recall the following estimate from Lemma 7. where ω = π 2 8π 2 g 2. E( X x (t, x)h 2 ) a 1 e 2ωt h 2 t, h, x, (76) Lemma 23. Let {ϕ n } Γ and let G L 2 (, µ; ) be such that lim Dϕ n = G n in L 2 (, µ; ). Then, for any t we have lim DR tϕ n = E[Xx(t, x)g(x(t, x)) n in L 2 (, µ; ). In particular, if Dϕ n in L 2 (, µ; ) we have DR t ϕ n in L 2 (, µ; ) for all t >. Proof. Write DR t ϕ n (x) = E[X x(t, x)dϕ n (X(t, x), t, x. Taking into account estimate (76) and the invariance of µ, yields DR t ϕ n (x) E[Xx(t, x)g(x(t, x)) 2 µ(dx) = E[X x(t, x)(dϕ n (X(t, x)) G(X(t, x)) 2 µ(dx) a 1 e 2ωt = a 1 e 2ωt = a 1 e 2ωt E[ Dϕ n (X(t, x)) G(X(t, x)) 2 µ(dx) R t ( Dϕ n G 2 )(x)µ(dx) Dϕ n (x) G(x) 2 µ(dx). The conclusion of the lemma follows.

26 26 Giuseppe Da Prato Proposition 24. D µ is closable. Moreover, if ϕ belongs to the domain of the closure D µ of D µ and D µ ϕ = we have that D µ R t ϕ = for any t >. Proof. Let {ϕ n } Γ and G L 2 (, µ; ) be such that By (73) we have that ϕ n in L 2 (, µ), Dϕ n G in L 2 (, µ; ). (R t ϕ n ) 2 dµ + t ds σdr s ϕ n 2 dµ = ϕ 2 n dµ. Letting n and taking into account that g is bounded below, yields lim n t ds DR s ϕ n 2 dµ =. Consequently, by Lemma 23, it follows that t ds (E[Xx(s, x)g(x(s, x)) 2 µ(dx) =, h. Then for almost all t we have that Now fix h. Then we have, E[ G(X(t, x)), h E[X x(t, x)g(x(t, x)) =. (77) E[ G(X(t, x)), X x (t, x) h + E[ G(X(t, x)), h X x (t, x) h = E[ G(X(t, x)), h X x (t, x) h. Taking into account the invariance of µ and (77), we find that R t ( G(x), h ) µ(dx) = E[ G(X(t, x)), h µ(dx) = [ E[ G(X(t, x)), h X x (t, x) h µ(dx) 1/2 [ 1/2 E[ G(X(t, x)) 2 µ(dx) E[ h X x (t, x) h µ(dx) 2. Therefore, as t we find by the strong continuity of R t in L 1 (, µ) G(x), h µ(dx) =

27 Kolmogorov equations for stochastic PDE s with multiplicative noise 27 and by the arbitrariness of h it follows that G = as required. Finally, the last statement follows from by Lemma 23. By Proposition 24 it follows that the mapping D µ : Γ L 2 (, µ) L 2 (, µ; ), ϕ Dϕ, is closable, let D µ its closure. We shall denote by W 1,2 (, µ) the domain of D µ and, if there is not possibility of confusion, we shall set D µ = D. Proposition 25. We have D(L µ ) W 1,2 (, µ) with continuous embedding. Moreover, the following identity holds L µ ϕ ϕ dµ = 1 σdϕ 2 dµ, ϕ D(L µ ). (78) 2 Proof. Let ϕ D(L µ ). Since Γ is a core for L µ, there exists a sequence {ϕ n } Γ such that By (72) it follows that σd(ϕ n ϕ m ) 2 dµ 2 ϕ n ϕ, L ϕ n L µ ϕ in L 2 (, µ). L (ϕ n ϕ m ) ϕ n ϕ m dµ. Therefore the sequence (Dϕ n ) is Cauchy in L 2 (, µ; ). Since D is closed it follows that ϕ W 1,2 (, µ) as required. 5.2 The Poincaré inequality Since 1/g is bounded, by Theorem 18 there is a unique invariant measure µ for R t and by the Doob theorem, see e.g. [12, we have that, lim R tϕ(x) = ϕ(y)µ(dy), x, (79) n for all ϕ C b (). Let us prove now the Poincaré inequality. Proposition 26. Assume that g π4. Then, for any ϕ W 1,2 (, ν) we have ϕ ϕ 2 dµ a 1 ω g 2 1/g 2 g(x)dϕ 2 dµ, (8) where ϕ = ϕdµ and ω = π2 8π 2 g 2.

28 28 Giuseppe Da Prato Proof. Let first ϕ Γ. Then by (73) we have R t ϕ(x) ϕ 2 µ(dx) = Moreover by (76) it follows that t E[ DR s ϕ(x) 2 E [ Dϕ(X(s, x)) 2 X x (s, x) 2 ds σdr s ϕ 2 dµ. (81) a 1 e 2ωs E [ Dϕ(X(s, x)) 2 = a 1 e 2ωs R s ( Dϕ 2 )(x). Taking into account (79) and the invariance of µ we obtain, R t ϕ(x) ϕ 2 µ(dx) + a 1 g 2 e 2ωs ds R s ( Dϕ 2 )(x)µ(dx) + a 1 g 2 1/g 2 e 2ωs ds g(x)dϕ(x) 2 µ(dx), and the conclusion follows. If ϕ W 1,2 (, µ), we proceed by density. Remark 27. If g is constant the condition g π4 is trivially fulfilled and we recover a result in [1. Remark 28. It is well known, see e.g. [8, that the Poincaré inequality implies that the spectrum σ(l µ ) of L µ consists of and a set included in the half-space {λ C : Re λ ω 1 }, (spectral gap) where ω 1 is a positive constant. The spectral gap in turn implies an exponential convergence of R t ϕ to the equilibrium R t ϕ ϕ 2 dν ce 2ω1t ϕ 2 dν, ϕ L 2 (R, ν),. (82) R where c is a suitable constant. R References 1. S. Albeverio and M. Röckner, Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab. Theory Relat. Fields, 89, , 1991.

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