Backward Ornstein-Uhlenbeck transition operators and mild solutions of non-autonomous Hamilton-Jacobi equations in Banach spaces

In this paper we revisit the mild-solution approach to second-order semi-linear PDEs of Hamilton-Jacobi type in infinite-dimensional spaces. We show that a well-known result on existence of mild solutions in Hilbert spaces can be easily extended to non-autonomous Hamilton-Jacobi equations in Banach spaces. The main tool is the regularizing property of Ornstein-Uhlenbeck transition evolution operators for stochastic Cauchy problems in Banach spaces with time-dependent coefficients


Introduction
Let E be a real Banach space and let T > 0 be fixed. The object of this paper is to study the existence of a mild solution V : [0, T ] × E → R to the following final-value problem for the non-autonomous semi-linear Hamilton-Jacobi partial differential equation ( (1.1) The final condition ϕ : E → R and the nonlinear Hamiltonian operator H : [0, T ] × E × E * → R are given, and for each t ∈ [0, T ], L t is the second-order differential operator Here ·, · denotes the duality pairing between E and its dual E * , {−A(t)} t∈[0,T ] is a family of densely defined closed linear operators generating a parabolic evolution family on E, {G(t)} t∈[0,T ] is a family of (possibly unbounded) linear operators from a Hilbert space H into E, Tr H [·] denotes the trace in H, and D x φ(x), D 2 x φ(x) denote first and second order Fréchet derivatives of φ : E → R at x ∈ D(A(t)).
In this paper we revisit the mild solution approach to Hamilton-Jacobi equations initiated by Da Prato [DP85] and Cannarsa [CDP91], and continued by Gozzi [Goz95,Goz96], Cerrai [Cer01a,Cer01b] and Masiero [Mas05] (see also Da Prato and Zabcyck [DPZ02], Zabcyck [Zab99] and the references therein). This approach consists in rewriting equation (1.1) in mild-integral form (cf. variation-of-constants formula) V (t, x) = [P (t, T )ϕ](x) + Here {W (t)} t∈[0,T ] is an H-cylindrical Wiener process defined on a probability space (Ω, F , P), E [·] denotes expectation in the Bochner-integral sense with respect to the probability measure P and B b (E) denotes the set of bounded Borel-measurable real-valued maps on E.
Under the so-called null-controllability condition (see Assumption A.2 in Section 5 below) the backward transition operators P (s, t) satisfy a strong regularizing property, see Theorem 5.8. For the case in which E is a Hilbert space and equation (1.1) is autonomous with respect to time variable (i.e. A(t) and G(t) do not depend on t), this regularizing property has been used in conjunction with a fixed point argument to prove existence of a unique solution to the integral equation (1.2) in a certain space of functions, see e.g. Theorem 9.3 in Zabcyck [Zab99, Sec. 9], Da Prato and Zabcyck [DPZ02, Part III] and Masiero [Mas05].
The main purpose of this paper is to show that this result can be easily generalized to the nonautonomous and Banach-space setting. Namely, we obtain the following (see Theorem 6.6 below) Theorem. Let ϕ ∈ C b (E). Suppose Assumptions (AT) and A.1-A.4 hold true. Then there exists an unique mild solution to equation (1.1).
We refer the reader to Sections 4-6 below for the precise statement of Assumptions (AT) and A.1-A.4. As an example, we consider a non-autonomous HJ equation in L p (0, 1) with p ≥ 2, see Example 6.7 below.
It should be emphasized that our proof does not present any significant innovation as we follow closely the arguments in the proof for the Hilbert-space case by Masiero [Mas05,Theorem 2.9]. However, to the best of our knowledge, this is the first paper that deals with infinite-dimensional non-autonomous semi-linear HJ equations in the general Banach-space framework, particularly in Lesbesgue spaces L p (O) with p ≥ 2. This is our main motivation to study HJ equations in a more general Banach-space setting that led to the writing of this paper.
The rest of the paper is organized as follows. In section 2 we recall some basic facts on Gaussian measures in Banach spaces, reproducing kernel Hilbert spaces and the Cameron-Martin formula. We present an alternative proof of a well-known result on regularizing property of Gaussian convolutions which first appeared in the seminal paper by L. Gross [Gro67]. In section 3 we review some results from van Neerven and Weis [vNW05a] on stochastic integration of deterministic operator valued functions with respect to a cylindrical Wiener process.
In section 4 we recall the setting of Acquistapace and Terreni for parabolic evolution families and non-autonomous evolution equations. In section 5 we introduce backward Ornstein-Uhlenbeck (OU) transition evolution operators in Banach spaces and extend some results from van Neerven [vN98, Section 1] on the relation between the associated reproducing Kernel Hilbert spaces. In section 6, we state and prove the final result Theorem 6.6. Throughout, as the main working example, we consider a linear parabolic second-order stochastic PDE with time-dependent coefficients and spacetime white noise formulated as an evolution equation in L p (0, 1) with p ≥ 2. We prove the transition operators of the (mild) solution verify the assumptions of the main result. This leads to our final Example 6.7.
Discussion. Of particular interest are Hamiltonians H of the form is the E-valued solution to the controlled non-autonomous stochastic evolution equation with additive noise For the case in which E is Hilbert, under certain additional differentiability assumptions on the Hamiltonian (1.4), the mild solution of (1.1) can be used to formulate optimality criteria and verificationtype results for optimal control problems in Hilbert spaces for stochastic PDEs, see e.g. Da Prato and Zabcyck [DPZ02, Part III] or Masiero [Mas05,. This can also be combined with Malliavin Calculus and backward stochastic evolution systems in Hilbert spaces to prove existence of an optimal feedback control, see e.g. Fuhrman and Tessitore [FT02a,FT02b,FT04a,FT04b] and the references therein.
Using regularizing properties of stochastic convolutions, Masiero [Mas08] proved existence of mild solutions of a certain class of autonomous HJB equations on the space of continuous functions C(O). Under additional, somewhat restrictive conditions on the nonlinear coefficient F, particularly a dissipative-type condition and a very specific form of dependence with respect to the control variable, Masiero also solved the control problem using backward SDEs but with no use of Malliavin calculus.
At the moment, we are unable to obtain optimality criteria and verification-type results for optimal control problems in Banach spaces for non-autonomous stochastic PDEs as this requires approximation results in C b (E) by smooth functions that do not seem available at the moment in the general Banach-space setting. However, we believe this can be overcome by employing recent results on Malliavin calculus in Banach spaces (see e.g. Maas [Maa10]). We will address this issue in a forthcoming paper.

Gaussian measures in Banach spaces, Cameron-Martin formula and smoothing property
We recall first some basic facts on Gaussian measures in Banach spaces, particularly the Cameron-Martin formula and the smoothing property of Gaussian convolutions in Banach spaces. Let B(E) denote the Borel σ−algebra on the real Banach space E, let E * be the continuous dual of E and let ·, · denote the duality pairing between E and E * .
If µ is a centered Gaussian measure on E, there exists an unique bounded linear operator Q ∈ L(E * , E) called the covariance operator of µ, such that for all x * , y * ∈ E * we have (see e.g. Bogachev [Bog98]). Notice that Q is positive in the sense that and symmetric in the sense that The Fourier transformμ of µ is given bŷ This identity implies that two centered Gaussian measures are equal whenever their covariance operators are equal. For any Q ∈ L(E * , E) positive and symmetric, the bilinear form on Q(E * ) given by is a well-defined inner product on Q(E * ). We denote with H Q the Hilbert space completion of Q(E * ) with respect to this inner product. The inclusion mapping from Q(E * ) into E is continuous with respect to the inner product [·, ·] HQ and extends uniquely to a bounded linear injection i Q : H Q ֒→ E.
Definition 2.2. The pair (i Q , H Q ) is called the reproducing kernel Hilbert space (RKHS) associated with Q.
It can be easily shown that the adjoint operator i * Q : E * → H Q satisfies i * Q x * = Qx * for all x * ∈ E * . Therefore, Q admits the factorization This factorization immediately implies that Q is weak * -to-weakly continuous and that, if E is separable, so is H Q . We identify for the sake of simplicity H Q with its image i Q (H Q ) ⊂ E.
We will denote with H µ (resp. i µ ) instead of H Q (resp. i Q ) whenever Q is the covariance operator of a Gaussian measure µ on E. In this case, we introduce a linear isometry from H µ into L 2 (E, µ) as follows: first observe that x * , · ∈ L 2 (E, µ) for every linear functional x * ∈ E * and that we have Here E µ denotes the expectation on the probability space (E, B(E), µ). Since Q is injective as an operator from E * into Q(E * ), the linear map is well-defined and is an isometry in view of (2.1).
Definition 2.4. We denote by the unique extension of the isometry (2.2) to H µ .
and this implies, in particular, that E µ [e iλ x * n ,· ] → E µ [e iλφµ(h) ] as n → ∞ for all λ ∈ R. Since x * n , · is normally distributed with mean 0 and variance |Qx * n | 2 Hµ , we have and by dominated convergence, taking the limit as n → ∞ we get Definition 2.5. For each h ∈ H µ we denote by µ h the image of the measure µ under the translation We call µ h the shift of the measure µ by the vector h.
Theorem 2.6 (Cameron-Martin formula). Let µ be a centered Gaussian measure on E with covariance operator Q ∈ L(E * , E) and let (i µ , H µ ) denote the RKHS associated with µ. Then, for any h ∈ H µ , the measure µ h is absolutely continuous with respect to µ and we have For the remainder of this section, we fix ϕ ∈ B b (E) and define the mapping ψ : E → R as The following regularizing property is a classical result proved by L. Gross in his seminal paper [Gro67, Proposition 9] using directly the notion of Fréchet derivative. Here we present an alternative proof based on Gâteaux differentiability.
Proposition 2.7. The map ψ : E → R is infinitely Fréchet differentiable in the direction of H µ . The first Fréchet derivative of ψ at x ∈ E in the direction of y ∈ H µ is given by and the second Fréchet derivative of ψ at x ∈ E in the directions y 1 , y 2 ∈ H µ is given by (2.5) Moreover we have the estimates Proof. Let us prove first that ψ is Gâteaux differentiable in the direction of H µ , i.e. that for all x ∈ E and y ∈ H µ , the mapping R ∋ α → ψ(x + αy) ∈ R is differentiable at α = 0. Let x ∈ E and y ∈ H µ be fixed and let α ∈ R. Observe that by the Cameron-Martin formula, we have is defined on a set E = E(y) of full µ-measure which depends only on y, for all α ∈ R. Thus, the mapping g : is well-defined and measurable. Moreover, for ε > 0 fixed we have the following estimate for all Hµ . (2.10) We know φ µ (y) is Gaussian random variable with moment generating function This implies, in particular, that exp(ε |φ µ (y)|) belongs to L 2 (E, µ). Since φ µ (y) ∈ L 2 (E, µ), by Hölder's inequality the right hand side in (2.10) belongs to L 1 (E, µ). Thus we may differentiate in the right hand-side of (2.8) with respect to α under the sign and obtain that the Gâteaux derivative of ψ at x in the direction of y is given by as well as the following estimate In turn this implies that the Gateaux derivative dψ : H µ → L(H µ , R) is continuous and uniformly bounded. Since ψ is also continuous and uniformly bounded on H µ , by Theorem 3 in Aronszajn [Aro76, Ch 2, Section 1] we conclude that ψ is Fréchet differentiable in the direction of H µ and (2.4) follows.
For the second-order Gâteaux derivative, if y 1 , y 2 ∈ H µ and α ∈ R we have where we have used again the Cameron-Martin formula and the change of variable ξ = z + αy 2 whose push-forward measure with respect with µ is given by µ αy2 .
Since both sides of (2.11) are continuous in y 1 ∈ H µ and Q(E * ) is dense in H µ , the above equality holds for any y 1 ∈ H µ . In addition, the equality holds for all ξ in a subset of E with full µ-measure that only depends on y 2 . Again, we can differentiate under the integral sign with respect to α to obtain the second Gâteaux derivative of ψ at x in the direction o y 1 and y 2 , together with the following estimate for all x ∈ E. By the same argument as above ψ is also twice Fréchet differentiable and (2.5) follows.
By identifying H µ with its dual H * µ , the map D 2 Hµ ψ(x) defines a bounded linear operator on H µ . The following lemma shows that it is actually a Hilbert-Schmidt operator. The proof follows the same argument as in the Hilbert-space case (see e.g. [DPZ02, Chapter 3]). We include the proof for the sake of completeness.
Hµ ψ(x) ∈ T 2 (H µ ) and Proof. Let (e i ) i be an orthonormal basis of H µ and let x ∈ E be fixed. Let us prove first the case ϕ ∈ C 1 b (E). By the same argument used in the proof of (2.4) one can derive Since the map φ µ is an isometry from H µ to L 2 (E, µ), the random variables φ µ (e k ), k ∈ N, form a complete orthonormal system in L 2 (E, µ) and by Parseval identity and dominated convergence we get and (2.13) follows. For the general case ϕ ∈ B b (E), we define the random variables Since φ µ (e k ), k ∈ N, are independent Gaussian random variables with mean 0 and variance 1, we get and where β(z) := ϕ(x + z). Thus, from the Parseval identity and Bessel inequality it follows that

Stochastic integration of deterministic operator-valued functions in Banach spaces
In this section we review some of the results from van Neerven and Weis [vNW05a] on stochastic integration of deterministic operator valued functions with respect to a cylindrical Wiener process. From this point onwards (H, [·, ·] H ) denotes a Hilbert space and (γ n ) n a sequence of real-valued standard Gaussian random variables on a probability space (Ω, F , P) endowed with a filtration F = {F t } t≥0 .
H , for all t ≥ 0 and y 1 , y 2 ∈ H, (ii) for each y ∈ H, the process {W (t)y} t≥0 is a standard one-dimensional Wiener process with respect to F.
Definition 3.2. R ∈ L(H, E) is γ−radonifying iff there exists an orthonormal basis (e n ) n≥1 of H such that the sum n≥1 γ n Re n converges in L 2 (Ω; E).
We denote by γ(H, E) the class of γ−radonifying operators from H into E, which can be proved to be a Banach space when equipped with the norm The above definition is independent of the choice of the orthonormal basis (e n ) n≥1 of H. Moreover, γ(H, E) is continuously embedded into L(H, E) and is an operator ideal in the sense that if H ′ and E ′ are Hilbert and Banach spaces respectively such that S 1 ∈ L(H ′ , H) and It can also be proved that R ∈ γ(H, E) iff RR * is the covariance operator of a centered Gaussian measure on B(E), and if E is a Hilbert space, then R ∈ γ(H, E) iff R is a Hilbert-Schmidt operator from H into E (see e.g. van Neerven [vN08] and the references therein). The γ−radonifying property in the following example goes back to Brzezniak [Brz96], and will be used later in our main Example 5.3. For the sake of completeness, we include a proof which follows closely arguments from van Neerven [vN08, Chapter 15].
Before we discuss the integral for L(H, E)−valued functions, we observe that we can integrate certain H−valued functions with respect to a H−cylindrical Wiener process W (·). For a step function of the form ψ = 1 (s,t] y with y ∈ H we define T 0 ψ(r) dW (r) := W (t)y − W (s)y.
This extends to arbitrary step functions ψ by linearity, and a standard computation shows that Since the set of step functions L 2 step (0, T ; H) is dense in L 2 (0, T ; H), the map extends to a (linear!) isometry from L 2 (0, T ; H) into L 2 (Ω). We now define the stochastic integral for deterministic L(H, E)−valued functions with respect to W (·).
if it belongs scalarly to L 2 (0, T ; H) and for all A ⊂ (0, T ) measurable there exists a random variable Y A ∈ L 2 (Ω, F , P; E) such that By Fernique's theorem, the E−valued random variables Y A are uniquely determined almost everywhere and Gaussian. In particular Y A ∈ L p (Ω; E) for all p ≥ 1.
For a function Φ : (0, T ) → L(H, E) that belongs scalarly to L 2 (0, T ; H) we define an operator Observe that I Φ is the adjoint of the operator If Φ(·)y is strongly measurable for all y ∈ H then R Φ maps L 2 (0, T ; H) into E. The following theorem characterizes the class of stochastically integrable functions with respect to the H−cylindrical Wiener process W (·). 2. There exists an E−valued random variable Y such that for all 4. There exist a separable Hilbert space H a linear bounded operator S ∈ γ(H, E) such that for all Moreover, for all y ∈ H the function Φ(·)y is stochastic integrable with respect to W (·)y and we have the series representation where (e n ) n≥1 is any orthonormal basis for H. The series converges P−a.s and in L p (Ω; E) for all p ∈ [0, ∞). The measure µ is the distribution of T 0 Φ(t) dW (t) and we have the isometry We conclude this section with a sufficient condition for stochastic integrability in spaces of type 2 (see e.g. van Neerven and Weis [vNW05a, Theorem 4.7] or [vNW05b, Theorem 5.1]).
Definition 3.6. E is said to be of type 2 iff there exists K 2 > 0 such that Proof. See Theorem 5.1 in van Neerven and Weis [vNW05b].

Parabolic evolution families
Since there is no unified theory for parabolic evolution families and non-autonomous evolution equations, we restrict in this paper to the class of parabolic problems and setting introduced by P. Acquistapace and B. Terreni in [AT87]. We start this section by recalling the definition of positive operators with bounded imaginary powers.
Definition 4.1. Let A be a closed, densely defined linear operator on a Banach space E. We say that A is positive if (−∞, 0] ⊂ ρ(A) and there exists C ≥ 1 such that Remark 4.2. It is well known that if A is a positive operator on E, then A admits (not necessarily bounded) fractional powers A z of any order z ∈ C (see e.g. Amann [Ama95, Chapter III, Section 4.6]). Recall that, in particular, for |Re z| ≤ 1 the fractional power A z is the closure of the linear mapping Moreover, if Re z ∈ (0, 1), then A −z ∈ L(E) and we have Definition 4.3. The class BIP(θ, E) of operators with bounded imaginary powers on E with parameter θ ∈ [0, π) is defined as the class of positive operators A on E with the property that A is ∈ L(E) for all s ∈ R and there exists a constant K > 0 such that For each t ∈ [0, T ] let A(t) be a densely defined closed linear operator on a Banach space E. For each s ∈ [0, T ], consider the following non-autonomous Cauchy problem (4.1) Definition 4.4. We say that y ∈ C((s, T ]; E) ∩ C 1 ((s, T ]; E) is a classical solution of (4.1) if y(t) ∈ D(A(t)) for all t ∈ (s, T ] and (4.1) holds.
Definition 4.5. We say that a classical solution y of (4.1) is also a strict solution if in addition y ∈ C 1 ([s, T ]; E), x ∈ D(A(s)) and A(t)y(t) → A(s)x as t → s.
Operators satisfying (AT1) are called sectorial (of type (φ, K, w)). Equation (4.1) is called parabolic because of the sectoriality of the operators A(t).
and the map y(t) = S(t, s)x is a strict solution of (4.1) for every x ∈ Y s . In this case we say that  Moreover, for all θ ∈ (0, µ) and x ∈ D (A(t) + wI) θ we have

Backward Ornstein-Uhlenbeck transition evolution operators
with moving time origin s ∈ [0, T ] and initial data x 0 ∈ E.
Definition 5.1. We say that an E−valued process Z(·) is a mild solution of (5.1) if for all (t, s) ∈ T the mapping S(t, s)G(s) has a continuous extension to a bounded operator from H into E, which we will also denote by S(t, s)G(s), such that the operator-valued function (s, t) ∋ r → S(t, r)G(r) ∈ L(H, E) is stochastically integrable on the interval (s, t) and We know from Theorem 3.5 that existence of a mild solution for (5.1) follows from the following condition Assumption A.1. For each (t, s) ∈ T the mapping S(t, s)G(s) : D(G) → E extends to a bounded linear operator from H into E, also denoted by S(t, s)G(s), such that the positive symmetric operator Q t,s ∈ L(E * , E) defined by Q t,s x * , y * := t s S(t, r)G(r)(S(t, r)G(r)) * x * , y * dr, x * , y * ∈ E * (5.2) is the covariance operator of a centered Gaussian measure µ t,s on E. where, for each t ∈ [0, T ], A t is the second-order differential operator with a, b, c ∈ C µ ([0, T ]; C([0, 1])) and a ∈ C ε ([0, 1]; C([0, T ])) for µ ∈ ( 1 4 , 1] and ε > 0 fixed. Assume further that inf t∈[0,T ],ξ∈[0,1] a(t, ξ) > 0.
For p ≥ 2 and t ∈ [0, T ], let A p (t) denote the realization in L p (0, 1) of A t with zero-Dirichlet boundary conditions, It is well-known that for w sufficiently large, the operator A p (·) + wI satisfies (AT) with parameters µ and ν = 1 (see e.g. Acquistapace and Terreni [AT87] or Tanabe [Tan79]). We will assume for simplicity and without loss of generality that w = 0.
Notice that G(t) is not a bounded operator unless p = 2. However, we can prove the following Example 5.3. For each (t, s) ∈ T the map S p (t, s)G(s) can be extended to bounded operator from L 2 (0, 1) into L p (0, 1) that is γ−radonifying and satisfies (5.3). Since L p (0, 1) has type-2 for p ≥ 2, from Example 5.2 it follows that Assumption A.1 holds for {S p (t, s)G(s)} (t,s)∈T with H = L 2 (0, 1) and E = L p (0, 1) with p ≥ 2. Equivalently, equation (5.4) has a mild solution in L p (0, 1).
We now introduce the transition evolution operators associated with the linearized equation (5.1). Suppose that Assumptions (AT) and A.1 are satisfied. Let B b (E) denote the set of Borel-measurable bounded real-valued functions on E.  ϕ(S(t, s)x + z) µ t,s (dz), x ∈ E, ϕ ∈ B b (E), (t, s) ∈ T Before we discuss the smoothing property of the OU transition operators, we extend to the nonautonomous framework some results by van Neerven [vN98, Section 1] on the relation between the spaces H t,s for different values of s < t. The first observation is the following algebraic relation between the operators Q t,s , which is immediate from their definition Q t,s = Q t,r + S(t, r)Q r,s S(t, r) * , 0 ≤ s < r < t.
The following is a direct consequence of Proposition 2.3, This combined with the identity S(t, r)Q r,s S(t, r) * = Q t,s − Q t,r , implies that S(t, r) maps the linear subspace Range Q r,s S(t, r) * of H r,s into H t,s . The next result shows that we actually have S(t, r)H r,s ⊂ H t,s .
This is well-defined since, by (5.6), if Q t,s x * = 0 then Q r,s S(t, r) * x * = 0. By (5.7) ψ y * extends to a bounded linear functional on H t,s with norm bounded by |Q r,s y * | Hr,s . Identifying ψ y * with an element of H t,s , for all x ∈ E * we have ψ y * , x * = [Q t,s x * , ψ y * ] Ht,s = Q r,s S(t, r) * x * , y * = S(t, r)Q r,s y * , x * .
Therefore, S(t, r)Q r,s y * = ψ y * ∈ H t,s and |S(t, r)Q r,s y * | Ht,s ≤ |Q r,s y * | Hr,s , and the desired result follows.
Next we characterize the equality of the Hilbert spaces H t,r and H t,s in terms of the restriction S(t, r) ∈ L(H r,s , H t,s ).
Theorem 5.7. For all 0 ≤ s < r < t we have H t,s = H t,r , as subsets of E, if and only if ||S(t, r)|| L(Hr,s,Ht,s) < 1.
Proof. We know already that H t,r ⊂ H t,s , so it remains to prove that H t,s ⊂ H t,r , if and only if ||S(t, r)|| L(Hr,s,Ht,s) < 1.
Finally, we establish the smoothing property of the Ornstein-Uhlenbeck transition operators. We need the following assumption, usually referred to as null-controllability condition (see Remark 5.9 below).
Notation. If condition (5.9) holds, we denote by Σ(t, s) the map S(t, s) regarded as an operator from E into H t,s . Notice that Σ(t, s) is bounded by the Closed-Graph Theorem, and we have S(t, s) = i t,s • Σ(t, s). As in Definition 2.4, let φ t,s : H t,s → L 2 (E, µ t,s ) denote the unique bounded extension of the isometry Q t,s (E * ) ∋ Q t,s x * → x * , · ∈ L 2 (E, µ t,s ).
Let C ∞ b (E) denote the set of infinitely Fréchet-differentiable real-valued functions on E. Using Proposition 2.7 together with the condition (5.9) we obtain the following The Fréchet derivative of the function P (s, t)ϕ : E → R at x ∈ E in the direction y ∈ E is given by and the second Fréchet derivative of P (s, t)ϕ at x ∈ E in the directions y 1 , y 2 ∈ E is given by In particular, we have the estimates Remark 5.9. The condition (5.9) has a well-known control theoretic interpretation: for each s ∈ [0, T ] consider the nonhomogeneous Cauchy problem with u ∈ L 2 (s, T ; H). The mild solution of (5.12) is defined as Then u ∈ L 2 (s, t; H) and we have t s S(t, r)G(r)u(r) dr = 1 t − s t s S(t, r)S(r, s)x dr = S(t, s)x that is, S(t, s)x ∈ H t,s according to (5.14), and Assumption A.2 follows. Moreover, by (5.15), we have 6 Mild solutions of Hamilton-Jacobi equations in Banach spaces (6.1) The final condition ϕ : E → R and the nonlinear Hamiltonian operator H : [0, T ] × E × E * → R are given, and for each t ∈ [0, T ], L t is the second-order differential operator  Observe that the trace term in (6.1) may not be well-defined since G(t) is not necessarily a bounded operator.
Definition 6.1. For α ∈ (0, 1), we denote with S T,α the set of bounded and measurable functions v : [0, T ] × E → R such that v(t, ·) ∈ C 1 b (E), for all t ∈ [0, T ), and the mapping is bounded and measurable.
The space S T,α is a Banach space endowed with the norm Definition 6.2. We will say that a function v : [0, T ] × E → R is a mild solution of the Hamilton-Jacobi equation (6.1) if v ∈ S T,α for some α ∈ (0, 1), for each (t, x) ∈ [0, T ] × E the mapping is integrable and v satisfies (6.2).
is continuous and bounded, and there exists C > 0 such that Example 6.4. If the Hamiltonian H has the form Remark 6.5. A Banach-space framework seems more suitable for certain control problems for stochastic PDEs. Consider for instance the following controlled stochastic PDE of reaction-diffusion type perturbed by additive space-time white noise on [0, T ] × (0, 1), X(t, 0) = X(t, 1) = 0, where, for each t ∈ [0, T ], A t denotes the second order differential operator introduced in Example 5.3. Stochastic PDEs of the form (6.5) feature when modeling the concentration, density or temperature of a certain substance under random perturbations. In these applications, it is useful to study running cost functionals that permit to regulate X(·) at some fixed points ζ 1 , . . . , ζ n ∈ (0, 1), say J(X, u) = E T 0 φ(t, X(t, ζ 1 ), . . . , X(t, ζ n ), u(t)) dt . (6.6) This running costs clearly requires that the solution X(t, ξ) is continuous with respect to the space variable ξ. Recently, Veraar [Ver10] (see also Veraar and Zimmerschied [VZ08]) have proved that weak solutions to the uncontrolled version of equation (6.5) exist and have trajectories almost surely in C([0, T ]; D(A p (0) δ )) for δ < 1 4 where A p (t) denotes the realization in L p (0, 1) of A t with zero-Dirichlet boundary conditions, see Example 5.3. If we choose p > 2 and δ ∈ 1 2p , 1 4 using Theorem 1.15.3 in Triebel [Tri78] and Sobolev's embedding theorem, it follows that is, cost functional (6.6) is now well-defined. This suggests to choose E = L p (0, 1) with p > 2 as state space for the above control problem and the associated Hamilton-Jacobi-Bellman equation (6.1).
As we mentioned in the introduction, at the moment we are unable to obtain optimality criteria and verification-type results for optimal control problems in Banach spaces for non-autonomous stochastic PDEs as this requires approximation results of in C b (E) by smooth functions that do not seem available at the moment in the general Banach-space setting. We will address this issue in a forthcoming paper.
We now present the final result of this paper, which generalizes to the non-autonomous Banachspace setting Theorem 9.3 in Zabcyck [Zab99] on existence of mild solutions to HJ equations in Hilbert spaces (see also Da Prato and Zabcyck [DPZ02, Part III] and Masiero [Mas05]).
Theorem 6.6. Let ϕ ∈ C b (E). Suppose Assumptions (AT) and A.1-A.4 hold true. Then there exists a unique mild solution to equation (6.1).
Proof. The argument is largely based on the proof of Theorem 2.9 in Masiero [Mas05]. For any v ∈ S T,α we define the function γ(v) by  and β > 0 is a parameter to be specified below. Let v 1 , v 2 ∈ S T,α .