A BSDEs approach to pathwise uniqueness for stochastic evolution equations

We prove strong well-posedness for a class of stochastic evolution equations in Hilbert spaces H when the drift term is Holder continuous. This class includes examples of semilinear stochastic damped wave equations which describe elastic systems with structural damping (for such equations even existence of solutions in the linear case is a delicate issue) and semilinear stochastic 3D heat equations. In the deterministic case, there are examples of non-uniqueness in our framework. Strong (or pathwise) uniqueness is restored by means of a suitable additive Wiener noise. The proof of uniqueness relies on the study of related systems of infinite dimensional forward-backward SDEs (FBSDEs). This is a different approach with respect to the well-known method based on the Ito formula and the associated Kolmogorov equation (the so-called Zvonkin transformation or Ito-Tanaka trick). We deal with approximating FBSDEs in which the linear part generates a group of bounded linear operators in H; such approximations depend on the type of SPDEs we are considering. We also prove Lipschitz dependence of solutions from their initial conditions.


Introduction 1.Problem and main result
In this paper we consider the problem of strong well-posedness for a class of stochastic partial differential equations (SPDEs) when the drift term is only Hölder continuous.
In a real and separable Hilbert space H we consider a stochastic evolution equation of the form where A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup (e tA ) t≥0 , the operator G : U → H is a bounded linear operator defined on another real and separable Hilbert space U and W is a cylindrical Wiener process on U (cf. Section 2).Moreover, C : [0, T ] × H → U is bounded and continuous and C(t, •) is β-Hölder continuous, uniformly in t ∈ [0, T ], β ∈ (0, 1).
In our framework there are examples of non uniqueness of solutions in the deterministic case (i.e., when W = 0 in (1.1)) (see [8] and [27]).Hence our main result on strong uniqueness, see Theorem 4.2, is due to the presence of the Wiener noise (regularization by noise).Clearly, such theorem holds also in the Lipschitz case β = 1 (in this case the result does not depend on W ).Moreover, the boundedness of C can be relaxed by a localization procedure (see Remark 4.3).Our pathwise result also implies the existence of a strong mild solution (cf.Remark 4.4).
Examples of singular semilinear SPDEs of the form (1.1) we can treat are stochastic damped equations which describe elastic systems with structural damping and semilinear stochastic 3D heat equations (see (1.2), (1.3) and Section 6 which is about applications).
In the literature the problem of regularization by noise for stochastic evolution equations (1.1) of parabolic type has been widely studied, see [8], [9], [10], [11], [19], [33] and the references therein.Moreover, we mention [4] for SDEs in Banach space, and [27] and [28], where the semilinear stochastic wave equation is studied.Usually in such papers pathwise uniqueness is obtained by using the Itô formula after solving a Kolmogorov equation.In finite dimension this is the so-called Zvonkin transformation or the Itô-Tanaka trick (see the seminal paper [31], the recent monograph [15] and the references therein).
Note that establishing the Itô formula in infinite dimensions is a delicate issue when the noise is cylindrical (cf.[8], [9], [10] and the references therein).We replace the previous approach with a method based on forward-backward SDEs (FBSDEs).This does not require proving the Itô formula, it works for hyperbolic and parabolic SPDEs and extends the method introduced in [27] for singular semilinear stochastic wave equations.
The techniques of [27] and [28] work only when the operator A appearing in equation (1.1) is the generator of a group of bounded linear operators.Here we introduce approximating FBSDEs in which the linear parts A n in the backward equation generate a group of bounded linear operators in H (cf. equation (1.6) below); such approximations depend on the type of SPDEs we are considering.Let us explain two examples of singular SPDEs of the form (1.1) we can consider.
The second class of equations we can treat is 3D semilinear stochastic heat equations like where we are dealing with the Laplace operator in H = U = L 2 ([0, π] d ) with periodic boundary conditions and G = (−∆) −γ/2 with γ ≥ 0. Such equations are also considered in [8,Example 6.1] with β-Hölder continuous drifts C (even with (−∆) −γ/2 C(X x t ) replaced by the more general term C(X x t )).However as the authors explain in [8] for β-Hölder continuous drift terms C they can only prove uniqueness in dimensions 1 and 2. On the other hand, in Section 6.3 we treat also the dimension d = 3.
As in [27] our method allows to establish Lipschitz dependence of solutions from their initial conditions (cf.Theorem 4.2): sup where X x1 and X x2 denote the weak mild solutions to (1.1) starting at x 1 and x 2 ∈ H, respectively.Estimates like (1.4) have not been proved before in papers on regularization by noise for stochastic evolution equations of parabolic type (cf.[8], [9], [10], [11], [4]).
Note that when C is only continuous and bounded even for parabolic SPDEs (with A which verifies Hypothesis 5.16) pathwise uniqueness for (1.1) for any initial x ∈ H is still an open problem (cf.[9], [11] and the references therein).

Strategy of the proof
First notice the particular structure of equation (1.1), which in the BSDE literature is referred to as structure condition: the drift belongs to the image of the diffusion operator G.Under the basic Hypothesis 2.1 the existence of a (unique in law) weak mild solution given by X t = e tA x + t 0 e (t−s)A G C(s, X s )ds + t 0 e (t−s)A GdW s , P-a.s., (1.5) t ∈ [0, T ], directly follows by an infinite dimensional version of the Girsanov theorem, see e.g.[29,Proposition 7.1] and [13,Section 10.3].
In order to prove pathwise uniqueness of solutions to equation (1.1), we complement equation (1.1) with a family of BSDE, which gives the following family of systems of FBSDEs −dY t,x,n τ = −A n Y t,x,n τ dτ + G C(τ, X t,x τ )dτ − Z t,x,n τ dW τ , τ ∈ [t, T ], with T ∈ (0, T ] and n ∈ N. Here, (A n ) n∈N is a sequence of linear operators on H and each A n generates a group of bounded operators (e tAn ) t∈R which pointwise approximates (e tA ) t≥0 .We stress that the idea of associating to (1.1) a BSDE has been exploited in [27] and [28], but in that case the assumption that A is the generator of a strongly continuous group (e tA ) t∈R is fundamental.In the present paper a crucial point is to consider a suitable sequence of approximating BSDEs, where if A is not the generator of a group of operators, each A n is.By an infinite dimensional version of Girsanov Theorem, the process is a cylindrical Wiener process on U up to time T .Under this transformation, system (1.6) reads as and this system has a unique solution (X t,x , Y t,x,n , Z t,x,n ) where (Y t,x,n , Z t,x,n ) is a pair of predictable processes belonging to L 2 (Ω; C([0, T ]; H)) × L 2 (Ω × [0, T ]; L 2 (U ; H)).Further, P-a.s.
Y t,x,n τ = e −(T −τ )An u T n (τ, Ξ t,x τ ), Z t,x,n τ = e −(T −τ ) ∇ G u T n (τ, Ξ t,x τ ), a.e.τ ∈ [t, T ], (1.8) where ∇ G denotes the Gâteaux derivative along the direction of G and u T n is the unique solution to Here, (R t ) t≥0 is the Ornstein-Uhlenbeck semigroup defined by R t [Φ](x) := E[Φ(Ξ 0,x t )], for any Φ ∈ B b (H; H), any t ≥ 0 and any x ∈ H (see Section 5.1), and Ξ 0,x τ is the Ornstein-Uhlenbeck process defined by means of (1.1) when C = 0.In mild formulation, the process Y t,x,n satisfies for every τ ∈ [0, T ].By setting t = τ = 0, by applying the operator e T An to the first and the last side of (1.10) and by taking (1.8) into account, we get for every T ∈ [0, T ].By replacing this formula in (1.5) it follows that for every t ∈ [0, T ].From there, to obtain (1.4) we need that the first integral in the right-hand side of (1.11) converges to 0 as n goes to +∞, and that the function u t n is smooth enough in order to get Lipschitz estimates of the addends which involve u t n in (1.11).Both these things are a consequence of Hypothesis 2.4, and so estimate (1.4) follows.

Plan of the paper
The paper is organized as follows.In Section 2 we state the main assumptions on the coefficients of equation (1.1), under which there exists a (unique in law) weak mild solution (X t ) to (1.1) (see Hypothesis 2.1).Further, we provide sufficient conditions which ensure existence and uniqueness of a smooth solution u T n to the integral equation (1.9) for any n ∈ N and any T ∈ (0, T ] (see Hypothesis 2.4).We stress that estimates on the Hilbert-Schmidt norm of ∇∇ G u T n , together with the family of systems of FBSDEs (1.6), are one of the main tool which we need to prove our result.Finally, we prove a generalized Gronwall Lemma which will be applied in the proof of Theorem 4.2.
In Section 3 we consider the family of systems of FBSDEs (1.10) and we show that, under our assumptions, for any n ∈ N, any T ∈ (0, T ], any x ∈ H and any t ∈ [0, T ) there exists a unique mild solution (X t,x , Y t,x,n , Z t,x,n ) to (1.7) which satisfies (1.8).
In Section 4 we prove the main result of the paper.At first we show that representation (1.11) holds true for the weak mild solution (X t ).Finally, by means of this representation and of Hypothesis 2.4, we prove Theorem 4.2, which states that there exists a positive constant c T such that for every x 1 , x 2 ∈ H estimate (1.4) is satisfied.
Section 5 is devoted to provide sufficient conditions on A, G and on C which ensure that Hypothesis 2.4 are verified.We split this section into three parts.In the former we show the existence of a unique smooth solution u T n to equation (1.9) for any n ∈ N and any T ∈ (0, T ], while in the second and in the latter we prove the crucial estimate on the Hilbert-Schmidt norm of ∇∇ G u T n when (A n ) n∈N are the Yosida approximants of A and when (A n ) n∈N are finite dimensional approximations of A, respectively.
Finally, Section 6 concerns with two concrete models to which our abstract results apply: where Λ : D(Λ) ⊂ U → U is a positive self-adjoint operator such that Λ −γ is trace class; (ii) a semilinear stochastic heat equation where ∆ is the Laplacian operator on L 2 ([0, π] d ) with periodic boundary conditions.
We note that 1. in the case of beam equation, the result is completely new; 2. in the case of the heat equation, the result extends the one from [8] to the dimension d = 3.
Appendix contains minimal energy estimates for the beam equation which are necessary for our approach.
To the best of our knowledge, these estimates are new and of independent interest.They are inspired by the spectral methods used in [1,22,30].

Notations
Throughout the paper we denote by H, K, U real and separable Hilbert spaces.L(H; K) denotes the Banach space of all bounded and linear operators from H into K endowed with the operator norm; we set L(H) = L(H; H).Moreover L 2 (H; K) ⊂ L(H; K) is the Hilbert space of all Hilbert-Schmidt operators, i.e., the space of operators where (e k ) is an orthonormal basis of H.We introduce the norm • L2(H;K) as We also set L 2 (H) = L 2 (H; H).
If F : H → K is Gâteaux differentiable at x ∈ H, we denote by ∇F (x) ∈ L(H; K) its Gâteaux derivative at x and by ∇ k F (x) its directional derivative in the direction of k ∈ H, i.e., Let G : U → H be a linear bounded operator.We are interested into differentiating along G-directions, i.e., directions k ∈ H such that k = Ga for some a ∈ U .We introduce the notation (1.12)

The abstract equation
We fix T > 0, and we will consider the following semilinear stochastic differential equation in the real separable Hilbert space H: where W = (W t ) is a cylindrical Wiener process on another real separable Hilbert space U .Recall that W is formally given by "W t = n≥1 W (n) t e n " where (W (n) ) n≥1 are independent real Wiener processes and (e n ) is a basis of U ; W defines a Wiener process on any Hilbert space U 1 ⊃ U with Hilbert-Schmidt embedding (cf.Section 4.1.2in [13] and Section 2 in [16]).Moreover A, C and G satisfy the following assumptions.
Hypothesis 2.1.(i) A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup (e tA ) t≥0 .
(ii) G : U → H is a bounded linear operator.
(iii) There exists α ∈ (0, 1/2) such that (iv) The function C : [0, T ] × H → U is bounded and continuous and, moreover, there exists a positive constant K and β ∈ (0, 1) such that Note that condition (2.2) implies that for any t > 0 the linear bounded operator 2) is implicitly assumed also in [8].It ensures that solutions have continuous paths with values in H (cf. Section 5.3 in [13]).Clearly, We underline that in Hypothesis 2.1 it is included the case U = H and G = I, if the stochastic convolution Q t , t > 0, is a trace class operator and (2.2) holds.
Recall that a (weak) mild solution to (2.1) is given by (Ω, F , (F τ ), P, W, X), where (Ω, F , (F τ ), P) is a stochastic basis on which it is defined a cylindrical U -valued F τ -Wiener process W and a continuous P-a.s.We say that equation (2.1) has a strong mild solution if, on every stochastic basis (Ω, F , (F t ), P ) on which there is defined a cylindrical F t -Wiener process W on U , there exists a weak mild solution.
To prove our main result we require the existence of a family of operators (A n ) n∈N , generators of C 0 -groups of linear operators ((e tAn ) t∈R ) n∈N , and for each n ∈ N we consider the integral equation (2.6) with A n in the place of A 0 , that is (2.7) Note that (2.7) is a mild formulation of a Kolmogorov PDE related to (2.1); on this aspect see Remark 2.10 which also compares such parabolic PDE with the one considered in [8].
We will denote by u T n the solution of this equation (2.7).On the family of operators (A n ) n∈N and on the solution of the above integral equation we make the following assumptions.(B) For each n ∈ N there exists a unique solution u T n to (2.7) which verifies: (i) there exists C T > 0, independent of n and T , such that (2.9) (ii) For any n ∈ N and any Further, we assume that there exists an integrable function h : (0, T ) → R + , independent of n ∈ N, such that for any n ∈ N we have for some integrable function h : (0, T ) → R + with h L 1 (0,T ) ≤ C for any T ∈ (0, T ], for some positive constant C = C(T ).
Note that for any x, y ∈ H and t ∈ (0, T ) we have (using an orthonormal basis (f k ) in U ) where u T n,j := u T n , e j (we are identifying L(U ; R) with U ).We will prove that the unique mild solution u T n of (2.7) satisfies (2.11), and to prove this estimate we will make use of (2.12).Remark 2.6.In the case of the wave equation as in [27] and [28] we can take A n = A and so e tAn = e tA because A is the generator of a group of operators, while in the case of the damped equation A n will be the Yosida approximations, i.e., where R(λ, A) is the resolvent operator of A, for any λ in the resolvent set of A. Finally, for the stochastic parabolic PDEs considered in [8] we will consider A n as the finite dimensional approximations of A.
We notice that the stochastic damped equation and the stochastic parabolic PDEs considered in [8] are reformulated as a stochastic evolution equation in H like (2.1), with A generator of a strongly continuous semigroup of linear operators, while the Yosida approximants of A and the finite dimensional approximations of A generate a group of linear operators.
We stress that if the drift C is Lipschitz continuous with respect to x, uniformly in t, existence of a strong mild solution follows in a standard way, see e.g.[13,Theorem 7.6].
Proposition 2.7.Assume that C : [0, T ] × U → H is continuous, bounded and h → C(τ, h) is Lipschitz continuous, uniformly with respect to τ .Then for any x ∈ H there exist a unique mild solution to equation (2.1).
To conclude this section, we provide the following generalization of the Gronwall lemma which will be used in the sequel (see [14,Lemma 3.1]).

and (ii) follows by (i).
Remark 2.9.We recall that under the assumptions of Lemma 2.8, if In the following remark we compare the Kolmogorov equations (2.7) used in the present paper and the ones considered in [8].
Remark 2.10.Let T ∈ (0, T ] and let n ∈ N. Formal computations give that the H-valued solution u T n of (2.7) formally solves the equation where , for any t ∈ [0, T ] and any x ∈ H, and f is a regular function.Kolmogorov equations similar to (2.13) are considered in [8] for the study of semilinear parabolic SPDEs (however, note that in equation ( 6) of [8] the term −e (T −t)An G C is replaced by G C).
On the other hand, the function v n := e −(T −t)An u T n formally solves which is similar to the equation (formally) solved by the function v considered in [27, Remark 6.2].

Ornstein-Uhlenbeck processes and approximated FBSDEs
In this section we consider a family of FBSDEs on the time interval [t, T ], with 0 ≤ t < T ≤ T .Namely in a complete probability space (Ω, F , P), for any n ∈ N and for τ ∈ [t, T ] we consider the following system of FBSDEs with forward and backward equations both taking values in H, given by The solution to the forward equation is the so called Ornstein-Uhenbeck process Ξ = (Ξ t,x τ ) and it is nothing else than equation (2.1) with drift C equal to 0. Moreover G, C and W are the same as in (2.1).Finally (A n ) n≥1 are given in Hypothesis 2.4, part (A).
Remark 3.1.We recall that under the general assumptions of the present paper we cannot consider the BSDE with A instead of A n because −A is not the generator of a semigroup of operators (see [17]).This case has been considered in [27] since A is the generator of a group of operators, and here we generalize the method introduced in [27] to the case of A generator of a semigroup of linear operators.
Let n ∈ N. Concerning the backward equation in the FBSDE (3.1), its precise meaning is given by its mild formulation: P-a.s. the pair of processes (Y t,x,n , Z t,x,n ) satisfies for any τ ∈ [t, T ] (cf.[16], [17], [21] and the references therein).Notice that in order to give sense to the BSDE in (3.1) we need that −A n is the generator of a C 0 -semigroup of bounded linear operators, and this is true if we assume that Hypothesis 2.4, part (A) is satisfied.
Concerning equation (3.2) recall that we endow (Ω, F , P) with the natural filtration (F W t ) of W (i.e., F W t is the smallest σ-algebra generated by W (n) s , n ≥ 1 and 0 ≤ s ≤ t) augmented in the usual way with the family of P-null sets of F .All the concepts of measurability, e.g.predictability, are referred to this filtration.
The solution of (3.2) will be a pair of processes is the Banach space of all predictable H-valued processes Y with continuous paths and such that and where C T ,n is a positive constant which depends also on T and n, and the map: Proof.Existence and uniqueness of a solution directly come from Lemma 2.1 and Proposition 2.1 in [21], that we can apply since C is bounded.Estimate (3.3) follows from [18], Remark 4.5, estimate (4.19).Since the process Ξ t,x is F W t,T -measurable (where F W t,T is the σ-algebra generated by W r − W t , r ∈ [t, T ], augmented with the P-null sets), it turns out that Y t,x t is measurable both with respect to F W t,T and F t ; it follows that Y t,x t is deterministic.
Next we prove an identification property that in the present paper we will apply to u T n , solution to equation (2.7), which is the analogous of the identification formulae proved e.g. in [16] for real valued functions (see also [26] for the case of functions defined on Banach spaces).Lemma 3.3.Let v : [0, T ] × H −→ H be a continuous function such that for every t ∈ [0, T ], v (t, •) is Gâteaux differentiable and the map (t, x) → ∇v (t, x) is Borel measurable.Let us fix (t, x) ∈ [0, T ] × H and let Ξ t,x be the Ornstein-Uhlenbeck process defined in Section 2. If ψ is a square integrable predictable process and then P-a.s, ∇ G v (τ, Ξ t,x τ ) = Zτ , for a.e.τ ∈ [0, T ].Proof.The result can be seen as an extension of [16,Proposition 5.6] to the case of an H-valued BSDE, and for this extension we use techniques similar to the ones in [27].Let ξ ∈ U and consider the real Wiener process (W ξ τ ) τ ≥0 , where and we study the joint quadratic variation between the real process v h (•, Ξ t,x ) and W ξ .Since we find Now we compute the joint quadratic variation in a different way, arguing as in [16,Lemmata 6.3 & 6.4] and we obtain that the real process v h (•, Ξ t,x ) admits joint quadratic variation with W ξ given by Comparing this formula with (3.5) we get that for a.e.s ∈ [0, T ] we have, P-a.s., Since H is separable, for any ξ ∈ U we get P-a.s.
The assertion now follows.
Next we are going to apply the previous Lemma 3.3 to u T n solution to equation (2.7).We assume that Hypothesis 2.4, part (B), holds true and recall that the operator A n generates a group of linear bounded operators (e σAn ) σ∈R .We want to show that the pair of processes satisfy the BSDE (3.2).This is the content of the following proposition.
Proposition 3.4.Assume Hypotheses 2.1 and 2.4 hold true.Let u T n be the unique solution to (2.7).Then the pair of processes defined in (3.6) is the unique solution of the BSDE (3.2).As a consequence, if the pair of processes Moreover, if a pair of processes (Y t,x,n , Z t,x,n ) is solution to (3.2), then by setting we get that and x ∈ H, is solution to equation (2.7), i.e., v n ≡ u T n .Proof.The arguments are similar to those used in the proof of the uniqueness part of [16,Theorem 6.2], and are adequated to this different context.For reader's convenience, we simply write u n instead of u T n .Starting from equation (2.7), we notice that for any τ ∈ [t, T ], Hence, denoting the conditional expectation we can rewrite (3.8) as . By the representation theorem for martingales, see e.g.[13], there exists ( Zt,x We conclude that the process u n (τ, Ξ t,x τ ), t ≤ τ ≤ T is a continuous semimartingale with canonical decomposition.By Lemma 3.3, we have that Zt,x Then the previous semimartingale reads as If we apply e −(T −τ )An to both the sides of this equation we get and by comparing this expression with (3.2), it is immediate to see that the pair of processes solves equation (3.2).
The "Moreover" part follows from the fact that, by Proposition 3.2, equation (3.2) admits a unique solution.

Pathwise uniqueness for the nonlinear SDE
Let's go back to the nonlinear SPDE (2.1), which we rewrite with initial time t = 0: The existence of a weak solution to (2.1) has been already discussed in Proposition 2.2.Let us set X x τ := X 0,x τ for any τ ∈ [0, T ] and x ∈ H.Then, P-a.s.we have Let τ ∈ [0, T ].In the next result, for any n ∈ N we consider u τ n , the regular solution of (2.7) with T = τ , whose properties are listed in Hypothesis 2.4, part (B).Proposition 4.1.Let Hypotheses 2.1 and 2.4 hold true.Then, for any n ∈ N and any τ ∈ [0, T ] we have Proof.Let 0 ≤ t ≤ τ ≤ T and x ∈ H.We consider a (weak) mild solution X t,x to . Such solution is defined on a stochastic basis (Ω, F , (F t ), P), on which it is defined a cylindrical F t -Wiener process W on U .
Let us set By the Girsanov theorem (see, for instance, [13, Section 10.3] or the Appendix in [9]) there exists a probability measure P = P τ on (Ω, F τ ), such that in (Ω, F τ , P) the process ( W σ ) σ is a cylindrical Wiener process up to time τ (it is not difficult to prove that P and P are equivalent).In (Ω, F τ , P) the process Since for OU stochastic equations pathwise uniqueness holds, we have, in particular, that (X t,x ) is a predictable process with respect to the completed natural filtration (F W s ) 0≤s≤τ generated by W .Let us consider the FBSDE system for any σ ∈ [0, τ ], a.e.. Hence, P-a.s. for any s ∈ [0, τ ], a.e., we have We recall that Let us consider now the initial time t = 0, and let us write (4.5) with σ = 0 ( we write X x instead of X 0,x ).We get for any τ ∈ [0, T ], and by applying e τ An to both sides, from (4.4) we deduce that, P-a.s., By replacing in (4.1) we infer that for any τ ∈ [0, T ], and the proof is finished.
The next result is our main uniqueness theorem.
Theorem 4.2.Let Hypotheses 2.1 and 2.4 hold true.Then, there exists a positive constant c = c(T ) > 0 such that for any x 1 , x 2 ∈ H we have where X x1 and X x2 denote (weak) mild solutions to (2.1) starting at x 1 and x 2 , respectively, and defined on the same stochastic basis.In particular, for equation (2.1) pathwise uniqueness holds.
Proof.Let x 1 , x 2 ∈ H and let us denote by X 1 and X 2 the mild solutions to (2.1) starting at x 1 and x 2 , respectively.We fix t ∈]0, T ].From (4.2) with τ = t, for any n ∈ N we have where Notice that in (4.8) we consider the function u t n such that u t n (t, •) ≡ 0. The crucial point is that estimates on u t n are uniform in t (cf.Hypothesis 2.4), part (B).We also note that x → e tA x is Lipschitz continuous with respect to x, uniformly with respect to t, and taking into account Hypothesis 2.4, part (B), point (i), we know that x → u t n (0, x) is Lipschitz continuous.For what concerns the stochastic integral, note that using Hypothesis 2.4, part (B), point (ii) with T = t, by the Itô isometry we find Using (4.9) in (4.8), for any n ∈ N we get where C T is a positive constant independent of n and t.Note that is a bounded function on [0, T ].Thus applying the generalized Gronwall lemma we infer where K T is a positive constant independent of n and t.We need to prove that Using the dominated convergence theorem, we get the assertion if we show that, P-a.s., Let us consider the first limit in (4.11) (the proof of the second limit is similar).Let us fix ω, P-a.s.; for any n ∈ N, we have This shows assertion (4.10) and completes the proof.
Remark 4.3.We point out that, using a localization argument as in [10] the boundeness of C can be dispensed.In particular, one can prove strong well-posedness of (1.1), for any x ∈ H, under Hypotheses 2.1 and 2.4 but replacing the condition on C with the weaker assumption: and moreover, for any ball B = B(z, r), z ∈ H, r > 0, the function C : [0, T ] × B → U is β-Hölder continuous, uniformly in t ∈ [0, T ] (the index β ∈ (0, 1) should be the same for any ball B but the Hölder norm may depend on the ball we consider).
Remark 4.4.By Theorem 4.2, using a generalization of the Yamada-Watanabe theorem (see [29] and [23]), one deduces that equation (1.1) has a unique strong mild solution, for any x ∈ H.
5 Analytic results on the associated Kolmogorov equation (2.7) In this section, assuming Hypothesis 2.1 we give sufficient conditions on A, G and C such that Hypothesis 2.4 is satisfied.This will imply the pathwise uniqueness result for equation (2.1) according to Theorem 4.2.
We split this section into two parts: in the former we provide preliminaries results on the equation (2.6) which involves a generator A 0 of a strongly continuous semigroup e tA0 on H; in the second part we will consider Hypothesis 2.4.

Preliminary results on equation (2.6)
Let us assume the following condition on the operator Q t , t > 0, introduced in (2.4).
Hypothesis 5.1 (Controllability).(i) For any t > 0 we have Im(e tA ) ⊆ Im(Q (ii) For any t > 0 we set Γ(t) := Q −1/2 t e tA : H → H. From (i) it follows that Γ(t) is a bounded linear operator for any t > 0. We assume that there exist a positive constant C = C T and measurable functions and (5.1) Remark 5.2.Since G ∈ L(U ; H) it follows that the best choice of Λ 1 and of Λ 2 in (5.1) and a suitable choice of It is well known that for any t > 0 condition Im(e tA ) ⊆ Im(Q t ) is related to the null-controllability of the abstract controlled equation In the sequel we will also need the following assumption.By the dominated convergence theorem the previous assumption implies that , with σ 1 > 0 and σ 2 ∈ (0, 1) satisfy hypotheses 5.1 and 5.
Further, with the condition β > max{0, (σ 1 ) −1 (σ 1 + σ 2 − 1)} the product We recall that (R t ) is the Ornstein-Uhlenbeck semigroup defined by R t [Φ](x) := E[Φ(Ξ 0,x t )], for any Φ ∈ B b (H; H), any t ≥ 0 and any x ∈ H.Under Hypothesis 5.1, from [13,Theorem 9.26] we infer that for any for any x, k ∈ H and any t > 0, where N (0, Q t ) is the Gaussian measure on H with mean 0 and covariance operator Q t (see for instance [12, Chapter 1]).Estimates (5.1) allow us to repeat verbatim the statements and the proofs of [27, Lemmata 4.1-4.3],and we collect these results in a unique Lemma.
Lemma 5.5.Let Hypotheses 2.1 and 5.1 hold true, and let R t be the Ornstein-Uhlenbeck operator defined in Section 2, acting on vector-valued functions.
(i) For any Φ ∈ B b (H; H) and any t > 0, the function x → R t [Φ](x) is Gâteaux differentiable and its Gâteaux derivative ∇R t [Φ](x) ∈ L(H; H) is given by (5.3).Moreover, for any T > 0 there exists a positive constant C = C T such that (5.5) and The last result we need follows from interpolation theory.Let β ∈ (0, 1).From [12, Theorem 2.3.
with equivalence of the norms (here, (X, Y ) β,∞ denotes the real interpolation space between the Banach spaces X and Y , for more details see [24]).Further, we denote by (R t ) the transition semigroup of the Ornstein Uhlenbeck process Ξ 0,x acting on Borel measurable real valued functions φ : H → R as There is a link between the H-valued Ornstein-Uhlenbeck transition semigroup (R t ) t≥0 and the scalar Ornstein-Uhlenbeck transition semigroup (R t ) t≥0 : for any Φ ∈ B b (H; H) and h ∈ H we set Φ h (x) := Φ(x), h H for any x ∈ H. From [8, Section 3] it follows that (5.10) With computations similar to the ones in the proof of [27,Lemma 4.4] we can prove the following result.For reader's convenience we provide a detailed proof in Appendix B.
We go back to the integral equation (2.6) and for any T ∈ [0, T ] we introduce the spaces E T 0 and E T 0,γ as follows.Definition 5.8.E T 0 is the space of functions u ∈ C b ([0, T ] × H; H) such that u(t, •) is Fréchet differentiable on H for any t ∈ [0, T ] and the map ∇u : [0, T ] × H → L(H, H) is strongly continuous and globally bounded.Moreover, for any k ∈ U and t ∈ For any γ ≥ 0, we set where u γ,T := sup It is easy to prove that E T 0,γ is a Banach space for any γ ≥ 0. Some further properties of functions u ∈ E T 0 are collected in the next remark.Remark 5.9.
To verify the previous inequality we write for x = y (we have to consider |x − y| ≤ 1 and |x − y| > 1) is Borel measurable on [0, T ] (with values in R + ).It is not difficult to prove the measurability of t → sup x∈H ∇ G u(t, x) L(U;H) .In order to show that we consider a countable dense subset D of {u ∈ U : |u| U = 1}.Let S be a countable dense subset of H.We note that by the continuity property of get assertion (5.17).
In the next result we will also use the Hölder continuity of C(t, •), t ∈ [0, T ].
Theorem 5.10.Let Hypotheses 2.1, 5.1 and 5.3 hold true and let A 0 be the generator of a strongly continuous semigroup e tA0 on H.
Then, there exists a unique solution u T to (2.6) in the sense of Definition 2.3 which belongs to E T 0 and there exists a positive constant M = M T which only depends on T , sup t∈[0,T ] e tA0 L(H) and sup s∈[0,T ] C(s, •) C β b (H;U) but not on T , such that u T 0,T ≤ M .Proof.Let us introduce the operator G defined on E T 0,γ by for any (t, x) ∈ [0, T ] × H, with γ > 0 to be chosen.We proceed in some steps.
Step I. We have to verify that G : E T 0,γ → E T 0,γ .We only check the more difficult part, i.e. we only verify that if u ∈ E T 0,γ , for a fixed y ∈ H, We will only prove that the other term ∇ y T t R s−t e (T −s)A0 G C(s, •) (x)ds can be treated in a similar way.
Step II.We claim that a suitable choice of γ implies that G is a contraction on E T 0,γ .For any u 1 , u 2 ∈ E T 0,γ we have to estimate the difference G u 1 − G u 2 γ,T .Let us only estimate the term since the other addends can be estimated in a similar way.We have, for (5.20) Since u 1 , u 2 ∈ E T 0,γ , the map x → ∇ G u i (s, x) C(s, x) is β-Hölder continuous from H into H, i = 1, 2, uniformly with respect to s ∈ [0, T ], and for any s ∈ [0, T ], with i = 1, 2. By applying (5.12) to (5.20) and taking into account (5.19) we get, uniformly in y and in (t, x) (see also (5.15)) where, from Hypothesis 5.3, C γ,T is a positive constant which goes to 0 as γ → +∞, uniformly with respect to T ∈ [0, T ].We get Similar arguments applied to the other terms of the norm Choosing γ large enough we deduce that G is a contraction on E T 0,γ and therefore it admits a unique fixed point u T .
Step III.Let us prove the last part of the statement.We will estimate the crucial term x ∈ H, k ∈ U, with |k| U = 1, since the other addends can be estimated in a similar way.We have, using (5.15), for any t ∈ [0, T ], Hence, starting from and arguing as before (using also that inf t∈(0,T ] Λ 2 (t) > 0) we arrive at where the function r → h(r) = Λ 1 (r) Using the previous estimate we can bound |∇ y ∇ G k u T (t, x)| H for any y ∈ H, |y| H ≤ 1, |k| U = 1, arguing as in (5.21).We obtain sup Arguing in a similar way, we obtain (cf.(A) in Hypothesis 2.4).Then, there exist unique solutions u T n to (2.7) which belong to E T 0 and there exists a positive constant M = M T , independent of n and T , such that u T n 0,T ≤ M .
Remark 5.12.Note that the previous result implies the validity of (2.9) in Hypothesis 2.4 for any choice of generators (A n ) verifying assertion (A) in Hypothesis 2.4.
In the next two sections we provide sufficient conditions for the validity of (2.10) in Hypothesis 2.4.We will consider two different types of approximations (A n ) n∈N for A: the Yosida approximations which we use to treat semilinear stochastic damped equations and the finite dimensional approximations which we use to deal with semilinear stochastic heat equations.

Sufficient conditions to ensure Hypothesis 2.4, using the Yosida approximations for A
Let us show that the solutions u T n of (2.7) satisfies (2.10) of Hypothesis 2.4 when for any n ∈ N. We notice that with this choice of A n Hypothesis 2.4, part (A), is fulfilled (cf Remark 5.12).We introduce the following additional assumption which will be verified in Section 6.2 for the damped equation.
Hypothesis 5.13.For any T > 0 there exists a positive constant C = C T such that where Λ 2 (t) is the function introduced in Hypothesis 5.1.
The first estimate of the Hilbert-Schmidt norm of u T n follows from the following lemma.
Lemma 5.14.Let Hypotheses 2.1, 5.1 and 5.13 hold true.Then: x ∈ H, t > 0, and for any T > 0 there exists a positive constant C = C T such that (5.25) and for any T > 0 there exists a positive constant C = C T such that (cf.Hypothesis 5.1)) (5.26) (iii) For any T > 0 there exists a positive constant C = C T > 0 such that for any β ∈ (0, 1) we have for any Φ ∈ C β b (H; H).Proof.Let T > 0 and let {e k : k ∈ N} be an orthonormal basis of U .From (5.3) and (5.24) we get for any Φ ∈ C b (H; H), any x ∈ H and any t ∈ (0, T ], and (i) follows.
To prove (ii) it is enough to consider (5.6) and (5.24), and to argue as in the proof of (i).
It remains to prove (iii).Analogous computations as for (5.12) in the proof of Lemma 5.6 (see Appendix B) give for any T > 0, anu β ∈ (0, 1) and any Φ ∈ C β b (H; H), where C = C T is a positive constant which only depends on T .To conclude, let us consider an orthonormal basis {e k : k ∈ N} of N. It follows that, see also the calculations (2.12) which gives the thesis.
In the next Theorem we investigate further properties of u T n , the solutions to (2.7) with A n = nAR(n, A); see Corollary 5.11.Note that (5.28) gives (2.11) with h = c.Theorem 5.15.Let Hypotheses 2.1, 5.1, 5.3 and 5.13 hold true, and let u T n be the solutions to (2.7) Further, for any t ∈ [0, T ] and any x, y ∈ H, the map x) belongs to L 2 (U ; H) and there exists a positive constant c = c(T ) which depends on T but neither on T nor on n such that (5.28) Proof.The fact that for each t ∈ follows by (i) in Lemma 5.14 taking into account that B(s, x) := ∇ G u(s, x) C(s, x) is a bounded continuous function on [0, T ] × H with values in H.
Arguing as in the first step of the proof of Theorem 5.10 one can show that From Theorem 5.10 and estimate (5.27) we infer (see also (5.15))

Sufficient conditions to ensure Hypothesis 2.4, using the finite dimensional approximations for A
Here we assume that U = H and Hypotheses 2.1, 5.1 and 5.3, where in particular it is assumed that C(t, •) ∈ C β b (H; H) for some 0 < β < 1 uniformly in t ∈ [0, T ] (see (2.3)).Moreover we require the following condition: Hypothesis 5.16.
1.A is self-adjoint, with compact resolvent, {e n : n ∈ N} is a complete orthonormal system in H which satisfies Ae n = −α n e n , with non-decreasing positive (α n ) n≥1 .
(i) Assumption (ii) in Hypothesis 5.3 is weaker than assumption 6 in [8] (such assumption 6 corresponds to the case when Λ 1 = Λ 2 ).Recall that, in general we have Λ 2 ≤ Λ 1 (see Remark 5.2).The main consequence of this fact is that our results apply to semilinear stochastic heat equation in dimension d = 3 (see Section 6.3), while examples in [8] only cover the cases d = 1 and d = 2.
Remark 5.18.In this section we consider the case when G is not necessarily a trace class operator and (5.30) holds true, since if G ∈ L 2 (H) then (5.24) is satisfied with U replaced by H. Indeed, for any and the estimate (5.28) follows at once.This implies that condition G ∈ L 2 (H) allows to get strong uniqueness by using Yosida approximations and the computations developed in Section 5.2, but for semilinear stochastic heat equation this does not lead to the sharp result.
Let n ∈ N. We consider E n := span{e 1 , ..., e n } the finite dimensional linear span generated by e 1 , ..., e n (see Hypothesis 5.16) and we let Π n be the projection of H onto E n : x, e k H e k . (5.31) As approximants of A we will consider in this section the finite dimensional truncations of A, given by Let us notice that this family of operators satisfies Hypothesis 2.4, part (A).For any T ∈ (0, T ] we consider the integral equation (2.7) which we rewrite here for the reader's convenience: We denote by u T n the solution of this equation (see Theorem 5.10).Following Remark 2.10 u T n solves where Indeed in mild formulation equation (5.34) can be rewritten as For every fixed n we let u T n,k := u T n , e k : [0, T ] × H → R its k-component, with k ∈ N. The following lemma states that for any n ∈ N we have u T n (t, x) ∈ E n for any (t, x) ∈ [0, T ] × H.
Lemma 5.19.Let u T n,k be as above.Then: (i) For any k = 1, . . ., n we have formula (5.36) follows.Further, for any j ≥ n + 1, we have From (5.5) and Remark 5.7 we infer that, for k ∈ H, |k| H = 1, The generalized Gronwall lemma 2.8 gives ∇ G u T n,j (t, x) = 0 for any t ∈ [0, T ] and any x ∈ H and from (5.37) we get (ii).
We notice that, up to revert time, equations (5.33) and (5.36) coincide with the mild integral equations ( 16) and ( 15) in the paper [8], respectively, with G and G k which are given here by We stress that from Corollary 5.11 we already know that u T n,k ∈ E T 0 and there exists a positive constant M = M T , independent of k = 1, . . ., n, T and n such that The next result gives a new estimate which is not present in the regularity results of Section 4 in [8].Indeed in such section estimates on the second derivatives of solutions are given using the operator norm; instead here we consider the stronger Hilbert-Schmidt norm.
Theorem 5.20.Let Hypotheses 2.1, 5.1, 5.3 and 5.16 be satisfied, and let and there exists h : (0, T ) → R + ∈ L 1 (0, T ), independent of n, such that for any n ∈ N and any t ∈ (0, T ) we have Further, there exists a positive constant C = C T , depending on Λ 1−β 1 Λ 2 L 1 (0,T ) , such that for any T ∈ (0, T ] we have Proof.Let T ∈ (0, T ] and n ≥ 1.Let us prove that ∇ G u T n belongs to B b ([0, T ] × H; L 2 (H)).We have Lemma 5.19).Arguing as in (2.12) we have This shows that, for any (t, x), the map: ) is a Hilbert-Schmidt operator from H into H.For any N ≥ 1, (t, x) ∈ [0, T ] × H, we introduce the approximating mappings: where Π n has been defined in (5.31).By the previous calculations and by Theorem 5.10 we deduce that for any (t, x) ∈ [0, T ] × H, we get the desired measurability property.
In the sequel C is a positive constant which may vary from line to line and which does not depend on n, k and T .In order to prove (5.39) we write as in (2.12) By Lemma 5.19 assertion (5.39) follows if we prove for any n ∈ N. Let us prove estimate (5.42).Arguing as in (5.15) we find, for any t ∈ [0, T ], Let us apply ∇ G to (5.36).We will take into account the regularizing properties of R t (see (5.5), (5.12) and Remark 5.7), and (5.43).
For any k ∈ {1, . . ., n}, define Using that inf t∈(0,T ] Λ 2 (t) > 0 and taking into account (5.36) we get . By applying the generalized Gronwall lemma 2.8 we infer for any t ∈ [0, T ] and k = 1, . . ., n, where ; by the integral equation verified by u T n,k we get where in the last passage we have used the definition (5.45).Moreover setting K T = e g L 1 (0,T ) and by (5.44) we have performed the second part of the last inequality.Therefore, Let us prove that for any k ∈ N the functions t → I k T (t) and t → J k T (t) only depend on T − t.Indeed, setting T − r = s in formula (5.45)where I k T (t) is defined we get and we see that and we see that J k T (t) depends only on T − t.Let us set then (5.39) is satisfied.It remains to prove (5.40).To this purpose note that, for each fixed k ≥ 1, I k and J k are bounded function on [0, T ] (uniformly in k).Indeed and similarly, with a constant We need more precise estimates of the L 1 -norms of (I k (T − t)) 2 and (J k (T − t)) 2 to get the estimate (5.42) on Using that I k and J k are uniformly bounded, we concentrate on giving bounds for I k L 1 (0,T ) and J k L 1 (0,T ) ; by the boundednes of I k and J k this will imply estimates for (I k ) 2 L 1 (0,T ) and (J k ) 2 L 1 (0,T ) .We have by the Fubini theorem which is the required dependence on α k .On the other hand, Hence, denoting by C 1 a constant depending on g L 1 (0,T ) we find Recalling the definition of h in (5.46), collecting the previous estimates it follows that The proof is complete.

Applications
In this section we consider two concrete models to which our results apply: stochastic semilinear damped Euler-Bernoulli beam equation and stochastic semilinear heat equations.

An example of stochastic damped Euler-Bernoulli beam equation
First we consider a nonlinear stochastic damped Euler-Bernoulli beam equation with the nonlocal term and hinged boundary conditions: with ρ > 0 and α ∈ [0, 1/2].Using the terminology of [5] this equation is in the class stochastic Euler-Bernoulli beam equations which describe elastic systems with structural damping.
Here, y 0 describes the initial position and y 1 the initial velocity of the particle, and Ẇ (τ, ξ) is a space-time white noise on [0, T ] × [0, 1] which describes external random forces.
In Section 6.2 we show that equation (6.1) can be reformulated in an abstract way as a stochastic evolution equation in a suitable space H of the form (1.1).
If the term c(•, •, •) satisfies the next conditions, then we are able to apply our results to equation (6.1) which give the pathwise uniqueness for mild solutions.
To deal with equation (6.1) first we have to show the well-posedness when c = 0, by proving that the stochastic convolution is well defined in H: this is an easy consequence of the fact that Λ −γ is a trace class operator on U .
Once that the well-posedness of the linear stochastic damped Euler-Bernoulli beam equation is proved, we investigate the regularizing effects of the associated transition semigroup (R t ) (cf.Section 5).These effects can be proved by means of optimal blow-up rates for the minimal energy associated to null controllability of related linear deterministic control systems.To this purpose we use a spectral approach to the damped elastic operators introduced in [5] and recovered in [22] and [30].For the stochastic damped Euler-Bernoulli beam equation we will get (cf.Hypotheses 5.1 and 5.13) for every γ ∈ (1/8, 1/4).So we are able to we prove that if and β given in (6.2) belongs to β, 1 where β = 8γ/(1 + 4γ), then pathwise uniqueness holds true for equation (6.1).

Setting and assumptions
We consider the following nonlinear stochastic damped beam equation which is a general form of (6.1): with ρ > 0 and α ∈ [0, 1/2] and γ ∈ (1/8, 1/4).Here, Λ : D(Λ) ⊂ U → U is a positive self-adjoint operator on a separable Hilbert space U such that Λ −2γ which is a trace class operator from U into U (6.4) and W = {W (τ ) : τ ≥ 0} is a cylindrical Wiener process on U .We aim at formulating this equation as an abstract stochastic evolution equation in the product space 3) we will assume that it is continuous and bounded and there exists a positive constant K and β ∈ (0, 1) such that (cf. (i) in Hypothesis 5.3).We will assume the following hypothesis.
Let (µ n ) be the family of eigenvalues of Λ.We are assuming that We also require Hypothesis 6.2.For any n ∈ N ρ 2 = 4µ 1−2α n .Remark 6.3.Concerning the basic example (6.1) in Section 6.1, we have , it follows that equation (6.3) can be reformulated as a stochastic evolution equation in H: which has the form of (2.1).
In the next Sections 6.2.2 and 6.2.3 we provide preliminary results.Then in Section 6.2.4 we prove that stochastic linear equation (6.9) or (6.3) with C ≡ 0 is well-posed in H.To this purpose we need Hypothesis 6.2 and condition (6.4).Finally, in section 6.2.5 we will formulate our main result on wellposedness for (6.3).

Spectral decomposition in H = U × U
Only in this subsection and in Appendix A we consider complexified spaces.We do not change the notation to not weigh down them.

Linear stochastic damped Euler-Bernoulli beam equations
Let us consider the problem where A α,ρ has been defined in (6.8).In the following we will refer to the solution of equation (6.18) as the Ornstein-Uhlenbeck process.We have α ∈ [0, 1) and ρ satisfying Hypothesis 6.2.
We will show that equation (6.18) is well-posed in H, i.e., for any x ∈ H, there exists a unique mild solution having continuous paths with values in H (cf. (2.5)).This is given by X t = e tAα,ρ x + t 0 e (t−s)Aα,ρ GdW s , t ∈ [0, T ], P − a.s.. (6.19)To this purpose we need to show that the stochastic convolution verifies condition (2.2) (cf.Section 5.3 in [13]).This is an easy consequence of (6.4).
The previous results show that we can apply Theorem 4.2 to equation (6.9).We finally get Theorem 6.7.Let A α,ρ and G be defined as in Section 6.2.1, and let the assumptions of Section 6.2.1 be satisfied.Then, for the nonlinear damped beam equation (6.9) the assertions of Theorem 4.2 hold.In particular, we have pathwise uniqueness for (6.9).
In the previous two cases we can apply Theorem 4.2 to equation (6.23) with C given in (6.25) and obtain pathwise uniqueness and Lipschitz dependence of initial conditions.Remark 6.8.Concerning (6.23) the main difference with [8] is that in such paper the authors require the assumption Hence, the best situation is β = 1 2 and above condition reads as According to (6.24) we need 1 + γ > d 2 . By combining this condition with γ < 1 3 it follows that the case d = 3 is not reached in [8].
We notice that a control u steers k to 0 at time t ∈ (0, T ] in H in (A.1) if and only if u steers k to 0 at time t ∈ (0, T ] in H in (A.2).Hence, the energy to steer k to 0 at time t in H, which is given by We will prove that (A.2) is null controllable and we provide an estimate to E C (T, k).
For any N ∈ N we set Clearly, ( GΛ −γ v 1 (t)) N → GΛ −γ v 1 (t) as N → +∞ in H. Further, from the previous decomposition we infer that ( GΛ −γ v 1 (t)) N ∈ D( A α,ρ ) for any N ∈ N and A α,ρ ( GΛ −γ v 1 (t)) N = φ T (t) Hence, recalling the definition of Φ + n and of Φ − n we get We notice that from (6.17) it follows that for any t ∈ (0, T ] for some positive constant M , it follows that A α,ρ (Gv 1 (t)) N converges to in H as N → +∞.Since A α,ρ is a closed operator, it follows that GΛ −γ v 1 (t) ∈ D( A α,ρ ), for any t ∈ [0, T ], and where we have integrated by parts under the second integral and we have used the fact that GΛ −γ v 1 (s) ∈ D( A α,ρ ) for any s ∈ [0, T ], and that Λ −γ v 1 (0) = Λ −γ v 1 (T ) = 0. We notice that for any s ∈ [0, T ] we have The mild formulation of the solution y to (A.2) implies that y(T ) =e T Aα,ρ (Ga) + T 0 e (T −s) Aα,ρ GΛ −γ v(s)ds = e T Aα,ρ (Ga) − e T Aα,ρ (Ga) = 0, which gives the claim.Now we estimate the L 2 -norm of the control v.We separately consider the two addends v 0 and v 1 .
As far as v 0 is concerned, from (A.

Hypothesis 2 . 4 .
(A) For any n ∈ N the operator A n generates a strongly continuous group of linear and bounded operators (e tAn ) t∈R ⊂ L(H) such that for any T > 0 we have sup t∈[0,T ] sup n≥1 e tAn L(H) = K T < ∞, lim n→∞ e tAn x = e tA x, x ∈ H, t ≥ 0. (2.8) which is continuous from [0, T ] with values in H.It is enough to prove that sup t≤T |δ 1 n (t)| H → 0 as n → ∞.We note that, for any compact set K ⊂ H, r ∈ [0, T ], there exists y = y K,r such that sup y∈K e rA − e rAn y = e rA − e rAn y K,r , and so lim n→∞ sup y∈K e rA − e rAn y H = 0. (4.12)Let us introduce the compact sets K t = {g(s)} s∈[0,t] ⊂ H, t ∈ [0, T ].From (4.12) it follows that |δ 1 n (t)| H ≤ t 0 sup y∈Kt e rA − e rAn y H dr ≤ T 0 sup y∈KT e rA − e rAn y H dr → 0, n → ∞.

. 29 )
where C 1 and C 2 are positive constants which depend on T , K T in (2.8) and sup s∈[0,T ] C(s, •) C β b (H;U) , but neither on n and T .Corollary 5.11 and (5.29) give the thesis.