Regularizing Properties for Transition Semigroups and Semilinear Parabolic Equations in Banach Spaces

We study regularizing properties for transition semigroups related to Ornstein Uhlenbeck processes with values in a Banach space E which is continuously and densely embedded in a real and separable Hilbert space H. Namely we study conditions under which the transition semigroup maps continuous and bounded functions into differentiable functions. Via a Gir-sanov type theorem such properties extend to perturbed Ornstein Uhlenbeck processes. We apply the results to solve in mild sense semilinear versions of Kolmogorov equations in E.


Introduction
In this paper we study transition semigroups, related to Ornstein Uhlenbeck and perturbed Ornstein Uhlenbeck processes with values in a Banach space E, that map continuous and bounded functions into differentiable functions.The Banach space E is continuously and densely embedded in a real and separable Hilbert space H. Similar properties have been extensively studied in relation to the strong Feller property of the semigroup, see e.g.(8).For such processes we also study a more general regularizing property, introduced in the paper (20) for Hilbert space valued processes.We apply these results to prove existence and uniqueness of a mild solution for a semilinear version of a Kolmogorov equation in E.
Namely, we study E-valued processes which are solutions of the stochastic differential equation where A is the generator of a semigroup of bounded linear operators e tA , t ≥ 0, which extends to a strongly continuous semigroup in H with generator denoted by A 0 .W is a cylindrical Wiener process in another real and separable Hilbert space Ξ, and G is a bounded linear operator from Ξ to H. Finally, we assume that for every ξ ∈ Ξ and for every t > 0, e tA Gξ ∈ E and e tA Gξ E ≤ ct −α ξ Ξ , for some constant c > 0 and 0 < α < 1 2 .The stochastic convolution W A (τ ) = τ 0 e (τ −s)A GdW s is well defined as an H valued Gaussian process when its covariance operator e sA 0 GG * e sA * 0 ds, τ ≥ 0, is a trace class operator on H; we assume further that the stochastic convolution W A (τ ) admits an E-continuous version.The map F : E → Ξ is continuous, Gateaux differentiable and F and its derivative have polynomial growth with respect to x ∈ E. Moreover GF : E → E is dissipative.When A, F and G satisfy these assumptions, there exists a unique mild solution, that is a predictable process X 0,x τ in E satisfying P-a.s. for τ ∈ [0, T ] , X 0,x τ = e τ A x + τ 0 e (τ −s)A GF X 0,x s ds + τ 0 e (τ −s)A GdW s .
Let us denote by Z 0,x t the Gaussian process in E associated to the case F = 0: Z 0,x t = e tA x + t 0 e (t−s)A GdW s .Let us denote by R t the transition semigroup associated to the process Z 0,x t .First we prove that R t maps continuous and bounded functions into Gateaux differentiable ones, if for 0 < t ≤ T e tA 0 (H) ⊂ Q 1/2 t (H) .Moreover, if there exists C t , depending on t, such that the operator norm satisfies

L(H,H)
≤ C t , for 0 < t ≤ T, then for every bounded and continuous function ϕ : E → R, This extends well known results in (8) for Hilbert space valued processes.By a Girsanov type theorem we are able to extend this property of the semigroup R t , to the transition semigroup P t , associated to the process X 0,x t in the case of F not equal to 0. A similar result is presented in the paper (2), for Hilbert space valued processes: in that paper the proof is achieved by means of the Malliavin calculus.We also remember that in the monograph (4) such a regularizing property for transition semigroups associated to reaction diffusion equations is achieved under weaker assumptions than ours, on the contrary in (4) no application to Kolmogorov equations is performed.
In analogy, we prove that if t (H) and if there exists a constant C t , depending on t, such that the operator norm satisfies then for every bounded and continuous function ϕ : E → R, R t [ϕ] is Gateaux differentiable in E in the directions selected by G.More precisely, for a map f : E → R the G-directional derivative ∇ G at a point x ∈ E in direction ξ ∈ Ξ is defined as follows, see (20): This definition makes sense for ξ in a dense subspace Ξ 0 ⊂ Ξ such that G (Ξ 0 ) ⊂ E, for more details see section 2. We prove that for every bounded and continuous function ϕ : E → R the G-derivative of R t [ϕ], satisfies, for every 0 < t ≤ T , Again by applying the Girsanov type theorem we are able to extend this property of the semigroup R t , to the transition semigroup P t .
One of the main motivations is to find a unique mild solution for second order partial differential equations of parabolic type in the Banach space E. Second order partial differential equations in infinite dimensions have been extensively studied in the literature: we cite the monograph (10) as a general reference, and the papers (14) and (20), where the non linear case is treated.In that book and those papers the notion of mild solution is considered and the second order partial differential equation is studied in an infinite dimensional Hilbert space.For partial differential equations of parabolic type on an infinite dimensional Hilbert space, also the notion of viscosity solution has been studied by many authors, we refer to (5) as a general reference, and to the fundamental papers (17) and (18).We also cite (15) and (16) where the notion of viscosity solution is introduced for Hamilton Jacobi Bellman equation related respectively to boundary stochastic optimal control problems and to stochastic optimal control of the Duncan-Mortensen-Zakai equation.On of the main motivation to study Hamilton Jacobi Bellman equations is to solve a stochastic optimal control problem.In this paper we study equations on a Banach space E of the following form: where A t is formally defined by A, F and G are the coefficients of the stochastic differential equation (1.1), ψ is a map from [0, T ] × E × R × E * with values in R and ϕ : E → R is bounded and continuous.By mild solution of (1.2) we mean a continuous function u : [0, T ] × E → R, Gateaux differentiable with respect to x for every fixed t > 0, satisfying the integral equation In order to find a unique mild solution we use a fixed point argument, so we have to impose some Lipschitz conditions on ψ, and a regularizing property for the semigroup P t : for every bounded and continuous function ϕ : E → R, we assume that ).We prove similar results for equations of the form In all the paper we will refer to equations (1.2) and (1.3) as semilinear Kolmogorov equations.Equation (1.3) has a less general structure than equation (1.2), but it can be solved in mild sense under weaker assumptions on the transition semigroup P t : we need only to require that for every bounded and continuous function ϕ : For what concerns already known results on partial differential equations in Banach spaces, we are not aware of any paper treating viscosity solutions for partial differential equations of parabolic type in Banach spaces; in the papers ( 6) and (7) viscosity solutions are presented for first order partial differential equations in Banach spaces.We also cite the paper (21), where second order partial differential equations in Banach spaces are solved in mild sense with a completely probabilistic approach: (21) generalizes the results obtained in (11), where Hamilton Jacobi Bellman equations in a Hilbert space are studied by means of backward stochastic differential equations.In (21) equations with the form of equation (1.3) are considered: ϕ and ψ are taken more regular than in the present paper, on the contrary no regularizing property on the transition semigroup is asked.In this paper we weaken the regularity assumptions on ψ and ϕ, see the discussion in section 5 for more details.
The paper is organized as follows: in section 2 we we study regularizing properties for Ornstein Uhlenbeck transition semigroups, in section 3 we present a Girsanov type theorem for a suitable perturbation of the Ornstein Uhlenbeck process, in section 4 we study regularizing properties for perturbed Ornstein Uhlenbeck processes: we want to stress the fact that to do this we cannot make a direct use of Malliavin calculus, as in ( 2), since we are working in Banach spaces.In section 5 we study mild solutions of semilinear Kolmogorov equations in Banach spaces and in section 6 we present some models where our results apply.

Regularizing properties for Ornstein Uhlenbeck semigroups
In the following, with the letter X and E we denote Banach spaces.L(X, E) denotes the space of bounded linear operators from X to E, endowed with the usual operator norm.If E is a Banach space we denote by E * its dual space.We denote by C b (X) the space of bounded and continuous functions from X to R endowed with the supremum norm.We denote by C 1 b (X) the space of bounded and continuous functions from X to R, with a bounded and continuous Fréchet derivative.With the letters Ξ and H we will always denote Hilbert spaces, with scalar product •, • .All Hilbert spaces are assumed to be real and separable.For maps acting among topological spaces, by measurability we mean Borel measurability.
From now on, let E be a real and separable Banach space and we assume that E admits a Schauder basis.Moreover E is continuously and densely embedded in a real and separable Hilbert space H.By the Kuratowski theorem, see e.g.(22), Chapter I, Theorem 3.9, it follows that E is a Borel set in H.
We are given a complete probability space (Ω, F, P) , and a filtration {F τ , τ ≥ 0} satisfying the usual conditions.For every T > 0, H p ([0, T ] , E) is the space of predictable processes (Y τ ) τ ∈[0,T ] with values in E, admitting a continuous version and such that The Ornstein-Uhlenbeck process is the solution in E to equation A is a linear operator in E with domain D (A), W is a cylindrical Wiener process in another real and separable Hilbert space Ξ, and G is a linear operator from Ξ to H. We will need the following assumptions.We refer to (19) for the definition of sectorial operator.
Hypothesis 2.1.We assume that either 1.A generates a C 0 semigroup in E; or 2. A is a sectorial operator in E.
In both cases we denote by e tA , t ≥ 0, the semigroup of bounded linear operators on E generated by A and we suppose that there exists ω ∈ R such that e tA L(E,E) ≤ e ωt , for all 0 ≤ t ≤ T .We assume that e tA , t ≥ 0, admits an extension to a C 0 semigroup of bounded linear operators in H, whose generator is denoted by A 0 or by A if no confusion is possible.
We have to make assumptions on G and on the stochastic convolution Let us introduce the nonnegative symmetric operators Q τ ∈ L (H, H) given by

The stochastic convolution W A (τ ) admits an E-continuous version.
It is well known that under assumption 1 alone, W A is a well defined Gaussian process in H and Q τ is the covariance operator of W A (τ ).Assumption 2 strengthens these properties.With hypotheses 2.1 and 2.2 there exists a unique mild solution (X x τ ) τ ∈[0,T ] to equation (2.1): by definition, where µ t (dy) is a symmetric gaussian measure in E. Indeed, due to hypothesis 2.2, the stochastic convolution W A is a Gaussian random variable with values in C ([0, T ] , E) with mean equal to e tA x, see e.g.(21).Let N e tA x, Q t (dy) denote the Gaussian measure in H with mean e tA x, and covariance operator Q t .
Lemma 2.3.The Gaussian measures µ t (dy) and N (0, Q t ) (dy) admits the same reproducing kernel, which is given by Q Proof.It is well known, see e.g.(8), that the reproducing kernel of the Gaussian measure N (0, Q t ) (dy) in H is given by Q 1/2 t (H).To conclude the proof it suffices to apply proposition 2.8 in (8).By this proposition if a separable Banach space E 1 is continuously and as a Borel set embedded in another separable Banach space E 2 , and if µ is a symmetric Gaussian measure on E 1 and E 2 , then the reproducing kernel spaces calculated with respect to E 1 and E 2 are the same.
In order to relate regularizing properties of the semigroup R t to the operators A and G we make the following assumptions: Hypothesis 2.4.Assume that for 0 < t ≤ T we have We denote by C t a constant such that Hypothesis 2.5.Assume that for 0 < t ≤ T we have We denote by C t a constant such that We note that if (2.2), respectively (2.3), holds then the operator e tA 0 G, is bounded by the closed graph theorem.
We recall that a map f : E → R is Gateaux differentiable at a point x ∈ E if f admits the directional derivative ∇f (x; e) in every directions e ∈ E and there exists a functional, the gradient ∇f (x) ∈ E * , such that ∇f (x; e) = ∇f (x) e. f is Gateaux differentiable on E if it is Gateaux differentiable at every point x ∈ E. We denote by G (E) the class of continuous functions f : E → R that are Gateaux differentiable on E and such that for every e ∈ E, ∇f (•) e is continuous from E to R.
Let G ∈ L (Ξ, H); we recall that for a continuous function f : H → R the G-directional derivative ∇ G at a point x ∈ E in direction ξ ∈ Ξ is defined as follows, see (20): A continuous function f is G-Gateaux differentiable at a point x ∈ H if f admits the Gdirectional derivative ∇ G f (x; ξ) in every directions ξ ∈ Ξ and there exists a functional, the We want to extend this definition to continuous functions f : E → R, where E ∈ H is a Banach space.In general, we can not guarantee that G (Ξ) ⊂ E. We make the following assumptions which is verified in most of the applications.
Hypothesis 2.6.There exists a subspace We say that a continuous function in every directions ξ ∈ Ξ 0 and there exists a linear operator where C x does not depend on ξ.So the operator ∇ G f (x) can be extended to the whole Ξ, and we denote this extension again by the class of bounded and continuous functions f : Lemma 2.8.Let hypotheses 2.1 and 2.2 hold true.
1.If hypothesis 2.4 holds true, then for every 2. If hypothesis 2.5 holds true, then for every Proof.We prove only 2, point 1 can be proved with similar arguments.We remark that when ξ ∈ Ξ, then Gξ ∈ H, but we cannot guarantee that Gξ ∈ E, anyway as a consequence of lemma 2.3, it turns out that Q 3) holds true, then e tA 0 G (Ξ) ⊂ E, so for every ξ ∈ Ξ, e tA 0 Gξ ∈ E for every t > 0. So, for every ξ ∈ Ξ, ∇ G (P t [ϕ]) (x) ξ is well defined by the formula Let ξ ∈ Ξ and consider the Radon-Nikodym derivative.Following (1), we denote by a µt the mean of µ t : for every f ∈ E * , a µt is defined as

It turns out that h ∈ H (µ t ) if and only if there exists g ∈ E *
µt such that h = R µt (g).By proposition 2.8 in (8), the reproducing kernel H (µ t ) coincides with Q 1/2 t (H).So by our assumptions se tA Gξ ∈ H (µ t ): there exists g ∈ E * µt such that e tA Gξ = R µt (g) and se tA Gξ = sR µt (g).
By the Cameron-Martin formula, see e.g. ( 1), corollary 2.4.3, We get where in the last passage we have used dominated convergence, since ϕ is bounded and But g ∈ E * µt , and so in particular We conclude that In the fourth passage we have used the fact that as a map from E * µt to H (µ t ), R µt is an isometric isomorphism, and in the last passage we have used the fact that the reproducing kernel of H (µ t ) is given by Im Q Remark 2.9.In the case of the Ornstein Uhlenbeck process, we are able to relate the assumption on the G-derivative of R t [ϕ] with properties of A and G. Also, in Hilbert spaces there are examples of Ornstein Uhlenbeck processes when it is clear that hypothesis 2.5 is less restrictive than 2.4, see (20).In particular we remember that, in the Hilbert space case, inclusion Im e tA ⊂ Im Q 1/2 t is equivalent to the strong Feller property of the Ornstein Uhlenbeck semigroup, see (8).

A Girsanov type theorem
We recall a result in (12) on a theorem of Girsanov type.Consider a stochastic differential equation with A and G satisfying hypotheses 2.1 and 2.2.Moreover we make the following assumption on b and G.
Here we prove an analogous result for a different perturbation of the Ornstein Uhlenbeck process Z x τ , the solution of equation (2.1).Let us consider a stochastic differential equation We make the following assumptions on the non linear term f .
We assume that as a map from E to Ξ, F is Gateaux differentiable and there exists j ≥ 0 such that F and its derivative satisfy the following inequalities, for every x, e ∈ E: Theorem 3.4.Assume that hypotheses 2.1, 2.2 and 3.3 hold true.Then for every ϕ ∈ C b (E) , Proof.We follow (12), theorem 1, and a simple idea well known already to Girsanov, see (13).We T and the sequence of stopping times τ n = inf t > 0 : 2 Ξ ds > n ∧ T .We define the probability measures P n (dω) = ρ x τn P (dω).The Novikov condition implies that is a cylindrical Wiener process under the probability P n .We claim that P n (τ n = T ) → 1 as n → ∞.In this case, in spite of the fact that Novikov condition cannot be applied directly, it is immediate that Eρ x T = 1: So the Theorem follows by the Girsanov theorem.We evaluate, by Markov inequality, With respect to the probability measure P n , Z x τ is solution to the equation and it follows that, on [0, τ n ], Y x τ,λ satisfies the equation So, on [0, τ n ], and for some satisfies the equation By the Gronwall lemma τn] (s) ds, and so for every p ≥ 1 where the last estimate follows from the fact that, as a process with values in C ([0, T ] , E), the stochastic convolution is a Gaussian process and so it has finite moments of every order.So the process (Y x τ ) τ , and consequently the process (Z x τ ) τ belongs to H p ([0, T ] , E) for every 1 ≤ p < ∞.By polynomial growth assumptions on G −1 f we get with C independent of n, and consequently 4 Regularizing properties of the semigroup: from the Ornstein Uhlenbeck semigroup to the perturbed Ornstein Uhlenbeck semigroup Let us consider the Ornstein Uhlenbeck process Z x τ which is a mild solution of the stochastic differential equation with values in E: In this section we assume that A and G satisfy hypotheses 2.1 and 2.2.Moreover we have to make one more assumption: as a consequence of lemma 2.3, it turns out that Q 2) or inclusion (2.3) hold true, then e tA 0 G (Ξ) ⊂ E. We have to make the following assumption on the norm of the operator e tA 0 G ∈ L (Ξ, E), for every t > 0.
Let us denote by R t the transition semigroup associated to Z x t , that is for every Moreover let us consider a perturbed Ornstein Uhlenbeck process X x τ which is a mild solution of the stochastic differential equation with values in E: Namely, a mild solution is an adapted and continuous E-valued process satisfying P-a.s. the integral equation From now on we assume that f satisfies hypothesis 3.3.From the proof of theorem 3.4, it follows that the process (X x τ ) τ belongs to H p ([0, T ] , E) for every 1 ≤ p < ∞.Let us denote by P t the transition semigroup associated to X x τ , that is for every ϕ ∈ C b (E) and t ∈ [0, T ] In this section we want to prove that if for every satisfies inequality (2.5), then also P t does.In order to prove these results we apply the Girsanov type Theorem 3.4 we have presented in the previous section: by this theorem we get and so P t can be written in terms of the expectation of a function of the process Z x .
and its G-derivative satisfies inequality (2.5), then also P t does.
Proof.Let η ∈ Ξ 0 .We want to prove that for every ϕ ∈ C b (E), ∇ G P t [ϕ] (x) η exists and satisfies the inequality So, since Ξ 0 is dense in Ξ, the linear operator ∇ G P t [ϕ] (x), which is well defined on Ξ 0 , admits an extension to the whole space Ξ, and we denote this extension again by We remark that if η ∈ Ξ 0 , Gη ∈ E and so the difference quotient First we prove that for every η ∈ Ξ 0 and for every ϕ ∈ C 1 b (E) By the definition of ∇ G P t [ϕ] (x) η we get Next we evaluate E ∇ϕ (X x t ) , e tA Gη .Let (ξ τ ) τ be a bounded predictable process with values in Ξ.We define X ε,x τ which is the mild solution to the equation By hypotheses 2.1, 2.2, 3.3 and 4.1 it turns out that there exists a unique mild solution; in particular by hypothesis 4.1 it turns out that τ 0 e (τ −s)A Gξ s ds, which a priori is an H-valued process, admits a version in C ([0, T ] , E) .
We define the probability measure Q ε such that Since X with respect to P and X ε with respect to Q ε have the same law, it turns out that where By differentiating with respect to ε, at ε = 0, and applying the dominated convergence theorem, we get where we have set that is

By hypothesis 4.1, it can be easily checked that
• X ξ τ is well defined as a process with values in the Banach space E. So for every bounded and predictable process (ξ τ ) τ we have proved Now we want to extend this equality to predictable Ξ-valued processes (ξ τ ) τ such that Ξ ds is finite.By hypothesis 4.1, also for such a process ξ, • X ξ τ is well defined with values in E and it is the unique mild solution of equation (4.3).Moreover for such a process ξ, there exists an increasing sequence ((ξ n τ ) τ ) n of bounded and predictable Ξ-valued processes such that Ξ ds → 0 a.s.We evaluate So we deduce that . By (4.4), for every ξ n bounded and predictable Letting n → ∞ we get for every predictable process ξ ∈ L 2 (Ω × [0, T ] , Ξ).

Now we look for a predictable process
Let us consider the deterministic controlled system where u ∈ L 2 ([0, T ] , Ξ).The solution of (4.5) is given by z s = s 0 e (s−r)A Gu r dr.
By hypothesis 4.1, the map s → z s is continuous with values in E. We claim that there exists ξ such that For such a process ξ, we get By Gronwall lemma We are looking for ξ ∈ Ξ such that t (H) and so there exists a control u ∈ L 2 ([0, T ] , Ξ) such that z t = e tA Gη.So for such a control u, by taking ξ s = u s − ∇F (X s ) z s , 0 < s < t, we get that .
By our assumptions, and so, by hypothesis 4.1, , is finite and for every η ∈ Ξ 0 it can be estimated in terms of η Ξ .It follows that E t 0 ∇F (X x s ) e sA Gη, dW s is finite and it can be estimated in terms of η Ξ .So Since the left hand side does not depend on the control u, on the right hand side we can take the infimum over all controls u steering, in the deterministic linear controlled system (4.5), the initial state 0 to e tA Gη in time t.The energy to steer 0 to e tA Gη in time t is given by E t, e tA Gη = min and it turns out that E t, e tA Gη = Q −1/2 t e tA 0 Gη .So for every ϕ ∈ C 1 b (E) and for every Since for every η ∈ Ξ 0 the norm of ∇ G P t [ϕ] (x) η is bounded by η Ξ , relation (4.6) can be extended to every η ∈ Ξ.It remains to prove that (4.6) can be extended to every ϕ ∈ C b (E).We extend the result in (24) valid for Hilbert spaces: by using Schauder basis the approximation performed in that paper can be achieved also in Banach spaces.So for every ϕ ∈ C b (E) there exists a sequence of functions ϕ n ∈ C 1 b (E) such that for every x ∈ E, ϕ n (x) → ϕ (x) as n → ∞, and sup x∈E |ϕ n (x)| ≤ sup x∈E |ϕ (x)|.So, by (4.6), we get for every x, y ∈ E Letting n tends to ∞ in the right hand side we get that from which it can be deduced the strong Feller property for the semigroup P t .We still have to prove that for every ϕ ∈ C b (E), P t [ϕ] is a G-Gateaux differentiable function on E. Let us consider the sequence of Frechet differentiable functions (ϕ n ) n that converges pointwise to ϕ in E. By previous calculations we get and the right hand side tends to 0 uniformly with respect to η ∈ Ξ, η Ξ ≤ 1.So there exists . By (4.7), for every η ∈ Ξ, the map x → H x η is continuous as a map from Ξ to R. By the estimate (4.6) we get that For every r > 0 and every η ∈ Ξ, we can write Letting n → ∞, we get If we divide both sides by r and we let r tend to 0, by dominated convergence and by the continuity of H x Gη with respect to x, we see that P t [ϕ] is G Gateaux differentiable and that ∇ G P t [ϕ] (x) Gη = H x Gη.Moreover the following estimate holds true: for every x ∈ E, η ∈ Ξ there exists a constant C > 0 such that

Applications to Kolmogorov equations
In this section we study semilinear Kolmogorov equations in the Banach space E.
Second order partial differential equations on Hilbert spaces have been extensively studied: see e.g. the monograph (10), and the papers ( 14) and (20), where the non linear case is treated, and (11), where it is not made any nondegeneracy assumption on G and such equations are treated via backward stochastic differential equations, BSDEs in the following.By means of the BSDE approach, in (11) Hamilton Jacobi Bellman equations of the following form are studied When G is not invertible, this a special case of an Hamilton Jacobi Bellman equation of the following form Moreover, in the BSDE approach, on the Hamiltonian ψ and on the final datum ϕ some differentiability assumptions are required, while in the approach followed e.g. by ( 14), (20) and also by the present paper, the Hamiltonian ψ and the final datum ϕ are asked lipschitz continuous; on the contrary some regularizing properties on the transition semigroup with generator given by the second order differential operator A t are needed.In the paper (21), which is a generalization of (11) to Banach spaces, the BSDE approach is used to solve second order partial differential equations in Banach spaces: also in ( 21) ϕ and ψ are taken Gateaux differentiable, and no regularizing assumptions on the transition semigroup are needed.In this paper we study Kolmogorov equations with the structure of equation 5.2 and under weaker assumptions on the transition semigroup related, with the structure of equation 5.1.One of the main motivation to study Hamilton Jacobi Bellman equations on a Banach space is to solve a stochastic optimal control problem related, which is well posed on a Banach space, for example a stochastic optimal control problem where the cost is well defined on a Banach space and/or a stochastic optimal control problem where the state evolves on a Banach space.This will be matter of a further research.
At first we study an equation of the following form where G ∈ L (Ξ, H), Ξ is another separable Hilbert space and ∇u (t, x) is the gradient of u.
Given f : E → E satisfying hypothesis 3.3 and the generator A of a semigroup on E, satisfying hypothesis 2.1, A t is formally defined by and it arises as the generator of the Markov process X in E, namely of the perturbed Ornstein Uhlenbeck process X t,x τ which is a mild solution of the following stochastic differential equation with values in E: x ∈ E. We denote by P t,τ the transition semigroup associated to X, that is for every To study equation ( 5.3) we also need the following assumptions on ψ and ϕ: R is Borel measurable and satisfies the following: 1. there exists a constant L > 0 such that We introduce the notion of mild solution of the non linear Kolmogorov equation (5.3).Since A t is (formally) the generator of P t,τ , the variation of constants formula for (5.3) is: (5.4) and we notice that this formula is meaningful if ψ (t, •, •, •) , u(t, •), ∇u (t, •) have polynomial growth, and provided they satisfy some measurability assumptions.We use this formula to give the notion of mild solution for the non linear Kolmogorov equation (5.3).
We introduce some function spaces where we seek the solution: for α ≥ 0, let C α ([0, T ] × E) be the linear space of continuous functions f : [0, T ) × E → R with the norm For α ≥ 0, we consider the linear space •) e is bounded and continuous as a function from [0, T ) × E with values in R. The space C s α ([0, T ] × E, E * ) turns out to be a Banach space if it is endowed with the norm Definition 5.3.Let α ∈ (0, 1).We say that a function u : [0, T ] × E → R is a mild solution of the non linear Kolmogorov equation (5.3) if the following are satisfied: We need the following fundamental assumption: Hypothesis 5.4.There exists α ∈ (0, 1) such that for every φ ∈ C b (E), the function P t,τ [φ] (x) is Gateaux differentiable with respect to x, for every 0 ≤ t < τ ≤ T .Moreover, for every e ∈ E, the function x → ∇P t,τ [φ] (x) e is continuous and there exists a constant c > 0 such that for every φ ∈ C b (H), for every ξ ∈ Ξ, and for 0 ≤ t < τ ≤ T , We want to stress the fact that condition (5.5) implies that the derivative blows up as τ tends to t and it is bounded with respect to x.In virtue of theorem 4. where α ∈ (0, 1) is given in hypothesis 5.4.
Proof.We give a sketch of the proof, that is similar to the proof of theorem 2.9 in (20).We define the operator Γ where α ∈ (0, 1) is given in hypothesis 5.4.Thanks to condition (5.5) Γ is well defined on C α ([0, T ] × E) × C s α ([0, T ] × E, E * ) with values in itself.We show that Γ is a contraction and so there exists a unique fixed point such that Γ (u, v) = (u, v).Since the gradient is a closed operator, Γ 2 [u, v] = ∇Γ 1 [u, v].We denote by (u, v) the unique fixed point of Γ, and v (t, x) = ∇u (t, x): u turns out to be the unique mild solution of equation (5.3).
Under less restrictive assumptions we can also study semilinear Kolmogorov equations in the Banach space E, with a more special structure.We study an equation of the following form where ∇ G u (t, x) is the G-gradient of u.
To study equation (5.8) we need the following assumptions on ψ: Hypothesis 5.6.The function ψ : [0, T ] × E × R × Ξ * → R is Borel measurable and satisfies the following: 1. there exists a constant L > 0 such that 3. there exists L ′ > 0 such that We introduce the notion of mild solution of the non linear Kolmogorov equation (5.8): again by the variation of constants formula for (5.8) is: and we notice that this formula is meaningful if ψ (t, •, •, •) , u(t, •), ∇ G u (t, •) have polynomial growth, and provided they satisfy some measurability assumptions.

Stochastic heat equations in bounded intervals
In (Ω, F, (F τ ) τ , P), we consider, for τ ∈ [0, T ] and ξ ∈ [0, 1], the following stochastic heat equation where Ẇ (τ, ξ) is a space-time white noise on [0, T ] × [0, 1], and h is a continuous function on [0, 1].Equation (6.1) can be formulated in an abstract way as where (W τ ) τ is a cylindrical Wiener process with values in H = Ξ = L 2 ([0, 1]).The operator A with domain D (A) is defined by It is well known that equation (6.1) admits a unique mild solution in H, given by since the stochastic convolution where g : R → R is a continuous and differentiable non increasing function, and g and its derivative have polynomial growth with respect to x.We can write this equation in an abstract setting: where for every x ∈ E, F (x) (ξ) = g (x (ξ)).F satisfies hypothesis 3.
holds true.We want to stress the fact that in the book (4) more general results about the strong Feller property for transition semigroups related to reaction diffusion equations are proved, on the contrary no applications to Kolmogorov equations in the space of continuous functions is given.
We consider the Kolmogorov equation relative to equation (6.4): where, at least formally, By estimate (6.5) and theorem 5.5, equation (6.6) admits a unique mild solution in E, if ψ and ϕ satisfy respectively hypotheses 5.1 and 5.2.
The initial datum h belongs to L 2 ρ (R).The choice of a weighted space is justified by the fact that we want to be able to treat constant initial functions.
In the space H = L 2 ρ (R) equations like (6.7) can be formulated in an abstract way as where (W τ ) τ is a cylindrical Wiener process with values in Ξ = L 2 (R) and J is the inclusion of Ξ in H.The operator A with domain D (A) is defined by It is well known, see e.g. ( 3) or ( 9), chapter 11, that equation (6.7) admits a unique mild solution in H, given by is well defined as a Gaussian process with values in H. Let ρ > 0: we consider the space C ρ (R) of all continuous functions on R such that e −ρ|ξ| |f (ξ)| → 0 as |ξ| → ∞.We consider the Banach space It turns out that E is continuously and densely embedded in H, and we want to prove that if the initial datum h ∈ E, then the process (Z τ ) τ is well defined as a process with values in E: to this aim we have to prove that the stochastic convolution W A (τ ) admits an E-continuous version.

First order stochastic differential equations
In (Ω, F, (F τ ) τ , P) let (β j (τ )) n j=1 be real, independent, standard Wiener processes.We consider, for τ ∈ [0, T ] and ξ ∈ [0, 1] the following stochastic equation where k j are integers such that k i = k j for i = j.We consider the Banach space E = {f ∈ C ([0, 1]) : f (0) = f (1)} which is continuously and densely embedded in the Hilbert space H = f ∈ L 2 ([0, 1]) .We define B : R n → E as the map that to (x 1 , ..., x n ) associates the function n j=1 e 2πik j (•) x j .We define A 0 by In the following, we write A instead of A 0 .
Equation (6.11) admits the abstract formulation where W τ = (β k 1 (τ ) , ..., β kn (τ )), so the space Ξ of the noise coincides with R n .If the initial datum h ∈ H, equation (6.12) admits a unique mild solution in H, moreover if the initial datum

Hypothesis 3 . 1 .
b : E → E is continuous and there exists an increasing function a : R + → R + with lim t→∞ a (t) = ∞ such that for every y ∈ E and x ∈ D (A), Ax + b (x + y) , x * E,E * ≤ a ( y ) − k x , for some k ≥ 0 and some x * ∈ ∂ x , the subdifferential of the norm of x.Moreover assume that G is invertible and thatG −1 L(H,Ξ) ≤ C.Denote by X x τ the solution of equation (3.1) and by Z x τ the solution of equation (2.1).Then the following result holds true, see(12), theorem 1. Theorem 3.2.Let A, b and G satisfy hypotheses 2.1, 2.2 and 3.1.Then, for every ϕ ∈ C b
We denote by H (µ t ) the reproducing kernel of the gaussian measure µ t .By lemma 2.3, it coincides with Q µ s t (dy) − E ϕ y + e tA x µ t (dy) , where µ s t (dy) = e tA sGξ + µ t (dy), and it is a gaussian measure on E with mean equal to e tA sGξ.t (H).Since by our assumptions e tA Gξ ∈ Q t (H), the gaussian measures µ s t (dy) and µ t (dy) are equivalent.Let us denote d (t, ξ, s, y) := dµ s t dµ t (y) 2, we know that if A and G satisfies hypotheses 2.1, 2.2, 2.4, with C τ −t = c/ (τ − t) α and 4.1, then hypothesis 5.4 is satisfied.
s)A dW s is well defined as a Gaussian process with values in H.Moreover, if the initial datum h ∈ E, then the process (Z τ ) τ is well defined as a process with values in E, in fact the stochastic convolution W A (τ ) admits an E-continuous version, see e.g.(9), theorem 5.2.9.E and e tA f E ≤ ct −1/4 f H .
Now let us consider a nonlinear heat equation in [0, 1]: for τ ∈ [0, T ] and ξ ∈ [0, 1] we consider the equation dx τ 3, see (8), example D.7.By (8), theorem 7.13, and (9), theorem 11.4.1, equation (6.4) admits a unique mild solution in E. Let us denote by P t the transition semigroup associated to equation (6.4).By theorem 4.2, for every ϕ ∈ C b (E) and every x, e ∈ E, P t [ϕ] (x) is Gateaux differentiable in any direction e ∈ E and the estimate |∇P t and this concludes the proof.So hypotheses 2.1 and 2.2 are verified for the coefficients of equation (6.8).Moreover hypothesis 2.5 holds true for the transition semigroup R t associated to equation (6.8), with C t = t = Ay t + Ju t,