On the equivalence of solutions for a class of stochastic evolution equations in a Banach space

We study a class of stochastic evolution equations in a Banach space $E$ driven by cylindrical Wiener process. Three different concept of solutions: generalised strong, weak and mild are defined and the conditions under which they are equivalent are given. We apply this result to prove existence, uniqueness and continuity of weak solutions to stochastic delay equation with additive noise. We also consider two examples of these equations in non-reflexive Banach spaces: a stochastic transport equation with delay and a stochastic McKendrick equation with delay.


Introduction
Let E be a Banach space and let H be a separable Hilbert space. In a given probability basis ((Ω, F, F, P), W H ), i.e. (Ω, F, P) is a complete probability space and W H is an H-cylindrical Wiener process with respect to a complete filtration F = (F t ) t≥0 on (Ω, F, P), consider the stochastic evolution equation: for initial condition Y 0 ∈ L 0 ((Ω, F 0 ); E). Here (A, D(A)) generates a strongly continuous semigroup (T (t)) t≥0 on a Banach spaceẼ such that D(A) ⊂ E ⊂ E are continuous and dense embeddings, and nonlinearities F : D(F ) ⊂ E → E and G : D(G) ⊂ E → L(H,Ẽ) are strongly measurable mappings with some regularity properties which we make precise in Sect. 2.
In the definition of a solution to the stochastic equation (SCP) one can either fix the probability basis ((Ω, F, F, P), W H ) on which the process Y lives in advance or make this to be part of a solution. In the former case the solution Y is usually called a stochastically strong solution, whereas in the latter case Y is a martingale or a weak solution in the probabilistic sense (stochastically weak solution). Here we only consider stochastically strong solutions of (SCP). For the notion of martingale solution of (SCP) in a Banach space see [18], where inter alia the equivalence between weak solutions in the probabilistic sense and local martingale problem is established.
We recall three analytical concepts of solutions to (SCP). We introduce the following definition of weak solution to (SCP) which is slightly more general than the one considered in [26] and [11].
In the following interpretation of solution to (SCP) we use the theory of stochastic integration in a Banach space as given in [25]. A process Y satisfying Definition 1.3 is called a analytically strong solution to (SCP) if in addition Y (t) ∈ D(A) a.s. for all t > 0 and AY is locally Bochner integrable (see [11]). The additional condition in the definition of analytically strong solution is not appropriate for stochastic delay equations (see Remark 4.10 in [7]) which we consider in Sect. 5, thus in this paper we do not focus on this concept of solution.
The equivalence of these three interpretations of solution to (SCP) in Hilbert space has been proved by Chojnowska-Michalik in [6] (see also [11,Theorem 6.5] and [27,Theorem 9.15]). For a linear (SCP) with additive noise in a Banach space the equivalence of weak and mild solution is given in [5], while in [22,Theorems 8.6 and 8.10] one can find the proof of equivalence of these three concepts of solution. In [29], the author considers mild, variational and weak solutions of non-autonomous stochastic Cauchy problems in a umd − Banach space. Applying the stochastic Fubini theorem he proves that mild and variational interpretations are identical. Moreover, only for reflexive Banach spaces, using Ito's formula, it is shown in [29] that weak and variational concepts are equivalent. In [7], the authors consider linear stochastic Cauchy problems in a umd − Banach space and formulate sufficient conditions for equivalence of mild and generalised strong solutions of (SCP) (see Theorem 3.2 in [7]). In the weak probabilistic setting and in a umd Banach space, assuming continuity of paths in the definitions of solutions and using localization and Itô's formula in the proofs, the equivalence between analytically weak and mild solutions is shown by Kunze in [18,Section 6].
In this paper we prove that the equivalence of Definitions 1.1-1.3 is also valid in umd − Banach spaces. Sections 3 and 4 show that in the fixed probability basis and without the assumption on the paths continuity three above-defined concepts of solutions to (SCP) are equivalent (Theorems 3.1, 4.1). These theorems are used in [7,Theorem 4.8] and in [16, Theorems 3.2, 3.6] to prove Markovian representation of stochastic delay equations in E × L p (−1, 0; E) for some p ≥ 1, where E is a type 2 umd Banach space. In Sect. 5 we apply the equivalence from Theorem 3.1 to prove the existence, uniqueness and continuity of weak solutions to a class of stochastic delay evolution equations in an arbitrary separable Banach space E (see Proposition 5.5).
It is worth mentioning that it turns out that for stochastic evolution equations with non-additive noise in a umd Banach space which are not type 2 it is convenient to analyse a concept of mild E η -solution of (SCP). This interpretation is more general than these considered in the article. The existence, uniqueness and Hölder regularity results of mild E η -solution to (SCP) with A being an analytic generator has been proved in [23]. Since the delay semigroup is not an analytic semigroup, we can not use these results in Sect. 5.
In the next section, mainly based on [25], we present sufficient conditions for the existence of stochastic integral in a umd − Banach space, and some preliminary lemmas which will be useful in the sequel.

Preliminaries
Here and subsequently, E,Ẽ stand for real Banach spaces, H denotes a real separable Hilbert space and W H is an H-cylindrical Wiener process on a given probability space (Ω, F, F, P). The following hypothesis will be assumed.
Banach spaceẼ such that T (t) ∈ L(E) for all t > 0 and D(A) ⊂ E ⊂Ẽ are continuous and dense embeddings.
In a typical example in which hypothesis (HO) is satisfied, E =Ẽ and (A, D(A)) generates a strongly continuous semigroup (T (t)) t≥0 on E (see also Example 3.2 in [18] ).
In the case whereẼ is not reflexive the adjoint semigroup (T * (t)) t≥0 is not necessary strongly continuous (cf. [13]). However sun dual semigroup (T (t)) t≥0 defined as subspace semigroup by [13] and Chapter 1 in [21]). A generator (A , D(A )) of the sun dual semigroup is given by Proof. Let n ≥ 1. By strong continuity D((A ) n ) is dense inẼ (see Proposition 1.8 in [13]), hence it is also * -weak dense inẼ . By Theorem 1.3.1 in [21] it follows thatẼ is * -weak dense inẼ * . Thus D((A ) n ) is * -weak dense in E * . The last property of D((A ) n ) gives the assertion of the lemma.
In the rest of this paper we assume the hypotheses.
such that for all t > 0 and x, y ∈ D(G) we have T (t)G(x) ∈ L(H, E) and Proof. In the case whereẼ = E is a Hilbert space see Lemma 9.13 in [27]. If (HO) holds, then we can repeat the reasoning from the proof of Lemma 9.13. Indeed, by strong continuity of (T (t)) t≥0 and (HA) there exists λ > 0 such that for all x ∈ D(F ) we have the following inequality where K E > 0 is a norm of continuous, linear embedding E →Ẽ. Since (A, D(A)) is closed and densely defined onẼ, The inequalities for G * may be handled in much the same way. Applying (HB) we conclude that for every x ∈ D(G) and all h ∈ H In the sequel we use the theory of stochastic integral for L(H, E)-valued process as introduced in [25]. For a Banach space with umd property one may characterise stochastic integrability in terms of γ-radonifying norm. umd property stands for Unconditional Martingale Difference property and it requires that all E-valued, L q (Ω; E)-convergent sequences of martingale differences are unconditionally convergent (see [15] and [25]). Throughout the paper, γ(H, E) stand for the space of γ-radonifying linear operators from H to E. The space γ(H, E) is defined to be the closure of the finite rank operators under the norm where the supremum is taken over all finite orthonormal systems h = (h j ) k j=1 in H and (γ j ) j≥1 is a sequence of independent standard Gaussian random variables. Hence γ(H, E) is a separable Banach space.
A H-strongly measurable, adapted process Ψ : [0, t]×Ω → L(H, E) such that Ψ * x * ∈ L 2 (0, t; H) holds a.s. for all x ∈ E * is stochastically integrable with respect to the cylindrical Wiener process W H if and only if Ψ represents γ(L 2 (0, t; H), E)-valued random variable R Ψ given by for every f ∈ L 2 (0, t; H) and for all x * ∈ E * (see [25,Theorem 5.9] ). For the sake of simplicity we shall say then that the process Ψ is in γ(L 2 (0, t; H); E) a.s. (see also Lemma 2.5, 2.7 and Remark 2.8 in [25]). If one wants a stochastic integral t 0 ΨdW H to be in L q (Ω; E) for some q > 1, then assuming that the process Ψ is scalarly in L q (Ω, L 2 (0, t; H)) (i.e. Ψ * x * ∈ L q (Ω; L 2 (0, t; H))) the L q -stochastic integrability of Ψ can be characterised by the existence of a random variable ξ ∈ L q (Ω; E) such that for all (2.5) Using decoupling inequalities (see [8]) one can prove the Burkholder-Gundy-Davies type inequality: for all q > 0. 1 In [15] it is shown that umd property can be characterised in terms of two properties: umd − and umd + .

Definition 2.3.
A Banach space E has umd − property, if for every 1 < q < ∞ there exists β − q > 0 such that for all E-valued sequences of L q -martingale differences (d n ) N n=1 and for all Rademacher sequences (r n ) N n=1 independent from (d n ) N n=1 we have the following inequality A Banach space E has umd + property, if the reverse inequality to (umd − ) holds. Recall that class of umd Banach spaces is in the class of reflexive spaces and includes Hilbert spaces and L q spaces for q ∈ (1, ∞). Moreover, class of umd − Banach spaces includes also non-reflexive L 1 spaces.
To integrate processes with values in umd − one needs a weaken notion of stochastic integral. In a Banach space E with umd − property the condition: Ψ is in γ(L 2 (0, t; H), E) a.s. is sufficient for the process Ψ to be stochastic integrable (cf. [24,Proposition 3.4] and [23,Section 8]). Moreover, if F is the augmented Brownian filtration F WH and Ψ * x * ∈ L q (Ω; L 2 (0, t; H)), and (2.5) holds for all x * ∈ E * , then it follows by the martingale representation theorem that Ψ is L q -stochastically integrable with respect to W H .

Equivalence of Weak and Mild Solutions
Theorem 3.1. In a umd − Banach space E consider (SCP) with hypotheses (HO), (HA) and (HB). Let Y be an E-valued strongly measurable, adapted process with almost surely locally Bochner square integrable trajectories. Assume that one of the following conditions holds (i) E is a umd space or F = F WH , and sup s∈[0,t] E Y (s) q E < ∞ for all t ∈ (0, ∞) and some q > 1; (ii) for all t > 0 the process:

Moreover, if there exists solution and the hypothesis (i) holds, then u →
Before proving the theorem, we formulate some remarks which are the consequences of Lemma 2.2 and the properties of stochastic integral in Banach spaces.
. Let Y be a E-valued, strongly measurable adapted process with locally Bochner square integrable trajectories a.s.
(i) Condition (HA) and Lemma 2.2 implies that E x → F (x), x * ∈ R is a Lipschitz-continuous function. Hence the condition (i) from the definition of weak solution to (SCP) is satisfied.

M. Górajski (ii) From (HB) and Lemma 2.2 it follows that E
x → G * (x)x * ∈ H is a Lipschitz-continuous function. Hence the process G * (Y )x * is strongly measurable and adapted with locally square integrable trajectories a.s.
are adapted, strongly and H-strongly measurable, respectively. Moreover, the first process has trajectories locally Bochner integrable a.s.
Proof of Theorem 3.1. We apply the stochastic Fubini theorem from [24] to obtain the key equations for the proof of Theorem 3.1: Eqs. (3.4), (3.5) and (3.7) below. As the process Y is assumed to be strongly measurable and adapted we may assume without loss of generality that E is separable. Moreover, observe that since every adapted and measurable process with values in Polish space has a progressive version, we may assume that Y is progressive.
Step 1. Fix x * ∈ D(A * ) and t > 0. Consider the processes: where s, u ∈ [0, t]. By Remark 3.2.(iii) it follows that Ψ 1 , Ψ 2 are H-strongly measurable. As Y is assumed to be progressive we conclude that for all s ∈ [0, t] and h ∈ H the selections: (Ψ 1 ) s h(u, ω) := Ψ 1 (s, u, ω)h, (Ψ 2 ) s h := Ψ 2 (s, u, ω)h are progressive. Hence by Proposition 2.2 in [25] we obtain strong measurability of Ψ * 1 x * , Ψ * 2 x * and for all s ∈ [0, t] progressive measurability of Ψ * 1 x * (s), Ψ * 2 x * (s). To apply the stochastic Fubini theorem (Theorem 3.5 in [24] Indeed, using Lemma 2.2 we have the following estimate On the Equivalence of Solutions for an Evolution Equations In the similar way we get Thus from stochastic Fubini's theorem it follows that By (3.6) and strong continuity of (T (t)) t≥0 it follows that for all x * ∈ D(A ) we have, almost surely, Step 2. Let us suppose that Y is a weak solution to (SCP), we prove that .7) and (3.4) and by the definition of a weak solution, we conclude that for all x * ∈ D(A ) and t > 0 one has, almost surely, Assuming that y * = A x * ∈ D(A ) and using the definition of a weak solution again, we can write the last term in (3.8) as follows where the first equality follows from the definition of a weak solution to (SCP) and in the last equality we use strong continuity of (T (t)) t≥0 and Fubini's theorem. Applying (3.9) into (3.8) we obtain, almost surely, On the Equivalence of Solutions for an Evolution Equations Hence for all x * ∈ D((A ) 2 ) one has, almost surely, Using the hypothesis (HO), (HA) and (HB) by the Krein-Smulyan theorem the above equality is also valid for all x * ∈ E * (see [18,Corollary 6.6] and the proof of Lemma 2.7 in [25]). Now we assume that hypothesis (i) holds and we prove that for all t > 0 the process u → Ψ(u) := T (t − u)G(Y (u)) is stochastically integrable on [0, t] with respect to W H . From the Step 1 of the proof it follows that the process Ψ is scalarly in L q (Ω; L 2 (0, t; H)). We define the random variable ξ Indeed, by the assumptions (HO), (HA) and (i) we obtain where in the last inequality we use the Minkowski integral inequality. From (3.10) it follows that ξ satisfies (2.5) for all x * ∈ E * . Hence, by Theorem 3.6 and Remark 3.8 in [25], the process Ψ is stochastically integrable on [0, t] with respect to W H and (ii) holds. Notice that by Lemma 2.1 the set D((A ) 2 ) separates the points ofẼ, thus also in E, and E is assumed to be separable, hence by the Hahn-Banach M. Górajski theorem there exists a sequence (x * n ) n≥1 of elements from D((A ) 2 ) which separates the points of E. Thus (3.10) holds simultaneously for all x * n on set of measure one. Therefore (1.1) holds.
On the other hand assume that Y is a mild solution to (SCP). By Remark 3.2.(ii) it follows that for all x * ∈ D(A * ) the process u → G * (Y (u))x * is stochastically integrable. Moreover, by (1.1) and then by Fubini's theorem and (3.7), and once more by (1.1) we obtain, almost surely, In a separable Banach space E let us consider the version of (SCP), where the noise is introduced additively i.e. G ∈ L(H,Ẽ). We will denote it by (SCPa). Here we do not need the assumption that E has umd − property, since stochastic Wiener integral in every Banach space is characterised by γ-norms (see [26,Theorem 4.2]). By Theorem 3.1 we obtain the following corollary.

then, Y is a weak solution to (SCPa) if and only if Y is a mild solution to (SCPa).
Moreover, if there exists a solution of (SCPa) and the hypothesis (i) holds, then u → T (t − u)G represents an element in γ(L 2 (0, t; H), E) for all t > 0.

Equivalence of Generalised Strong, Weak and Mild Solutions
In [7] a generalised strong solution to (SCP) is defined and its equivalence to a mild solution of (SCP) is proven. Under weaker assumptions we establish in   is also γ-bounded as an integral mean of γ-bounded operators (see Theorem 9.7 in [22]). Hence using the multiplier theorem due to Kalton and Weis [17] we obtain: if G(Y (·)) is in γ(L 2 (0, t; H), E) a.s., then the processes M. Górajski 0, t; H), E) a.s. (iii) LetẼ be a reflexive Banach space, and for all t > 0 the process G(Y (·)) is in γ (L 2 (0, t; H), E) a.s. Assume that and (HA'), (HB) are satisfied and Y has almost all trajectories locally square integrable. Then, it is easy to prove that a weak solution to (SCP) is a generalised strong solution to (SCP). Indeed, let Y be a weak solution to (SCP), then for every t > 0 and all x * ∈ D(A * ) we have the equality ds ∈ E ⊂Ẽ =Ẽ * * . By reflexivity ofẼ it follows that D(A * ) is dense inẼ * , hence, almost surely, the right hand side of (4.2) has an extension to bounded linear functional onẼ * . Thus by the definition of A * one has Fix t > 0. Let Y be a mild solution of (SCP) satisfying the assumptions of Theorem 4.1. Observe that [0, t] × Ω (u, ω) → t 0 Ψ 1 (s, u, ω)ds, where Ψ 1 is a process defined in Step 1 of the proof of Theorem 3.1, satisfies the assumptions of Lemma 2.8 in [7], i.e. for all h ∈ H process Φ 1 (s)h ∈ D(A) a.s. and the processes Moreover, by (HA') and the properties of strongly continuous semigroup we obtain, almost surely,

Existence, Uniqueness and Continuity of Solutions to Stochastic Delay Equations
In this section we apply the results from Sections 2, 3 to establish the existence of a unique continuous solution to a stochastic delay evolution equation of the form: where (B, D(B)) generates a semigroup of linear operators (S(t)) t≥0 on a separable Banach spaceẼ, X t : Ω × [−1, 0] → E is a segment process defined as X t (θ) = X(t + θ), θ ∈ [−1, 0], and E is a separable Banach space such that the hypothesis (HO) holds with A := B. We will use the following assumptions on mappings φ and ψ: and p ≥ 1, is densely defined mapping and there exists a ∈ L p loc (0, ∞) such that for all t > 0 and X , In [7] and [16] the Markovian representation of stochastic delay evolution equations with state dependent noise (i.e. ψ := ψ(X(t), X t )) in type 2 umd Banach spaces is proven. Using the same arguments we obtain the following representation for (5.1).

(i) If X is a weak solution to (5.1), then the process Y defined by Y (t) := [π 1 Y (t), π 2 Y (t)] = [X(t), X t ] is a weak solution to a stochastic evolution equation in
for all s ≥ 0 and all x ∈Ẽ (cf. Theorem 3.25 in [3]). Proof. The part (i) may be proved in much the same way as the corresponding part of Theorem 3.9 in [16].

(ii) Assume that the hypotheses (Hφ) and (HO) with
For the proof of the second part it is enough to show that weak and mild solutions to (5.3) are equivalent (see the proof of Theorem 3.9 in [16]). Using the Lemma 3.1 in [16] one can prove that the condition (HA) holds, hence we can apply the Corollary 3.3 to obtain the desired equivalence of solutions.
The delay equations play a crucial role in modelling phenomena e.g. in bioscience (cf. [1], [14]) economics and finance ( [19], [20]). Here we consider delay evolution equation with the Wiener additive noise, for delay equation in umd type 2 Banach space with more general Wiener noises we refer to [7], [16], where the reader can also find a more extensive literature overview. For stochastic delay evolution equation with infinite delay see [9]. At the end of this section we give two examples of stochastic delay partial differential equation in non-reflexive Banach space. First, in C 0 ([0, 1]) we examine a simple stochastic delay advection-reaction equation. This equation can be used to model product goodwill (see [2] for deterministic goodwill model without delays). In the second example in the state space L 1 (0, ∞) we analyse a stochastic delay age-dependent equation of the Sharpe-Lotka-McKendrick (or von Foerster) type (see [31] and [28]).
In Sect. 5.1 we recall the existence and continuity results for stochastic evolution equations with additive noise in separable Banach spaces.

Moreover, if the solution Y exists then there exists L > 0 such that for all
x, y ∈ L q (Ω; E) and s ≥ 0: the probability distribution of Y (s; x) does not depend on cylindrical Wiener process W H and the underlying probability space.
Proof. The implication (1)⇒(2) follows from the second part of Corollary 3.3. For the proof of implication (2)⇒(1) let us fix q ≥ 1, t > 0 and y ∈ L q ((Ω, F 0 ); E) and assume (2). By Remark 3.4 the condition (2) holds for all t > 0. We define a mapping K by for all Z ∈ SL q F (0, t; E). The symbol SL q F (0, t; E) stands for the Banach space of strongly measurable, adapted process Y with the Bilecki's type norm q for some β > 0. By assumptions, it follows that both stochastic and Bochner integrals in the definition of K are well defined. The first term of K is continuous a.s. Corollary 6.5 in [26] yields that the stochastic convolution in K is a continuous process in q-th moment. Moreover, by (HA) and the Minkowski's integral inequality for all Z ∈ SL q F (0, t; E) and for every s ∈ [0, t] one gets On the Equivalence of Solutions for an Evolution Equations (u)e −βu du. Between the same lines using (HA) for all Hence for β > 0 large enough the operator K is a strict contraction in SL q F (0, t; E). Therefore, the existence and uniqueness results follows by the Banach fixed-point theorem and by Corollary 3.3. Analysis similar to that in the proof of Theorem 9.29 in [27] shows that the second part of theorem is true.
Using the factorization method as introduced in Sect. 3 of [10] and Theorem 3.4 in [7] (see also Theorem 3.3 in [30]) we obtain sufficient condition for continuity of a solution to (SCPa).

Stochastic Delay Evolution Equation
Using Theorems 5.3 and 5.4 (see also Corollaries 3.12 and 3.13 in [16]) we obtain the proposition.  Proof. In the proof we use Theorem 5.2 and then we apply Theorems 5.3 and 5.4 to problem (5.3). Let t > 0. We shall now check the assumptions of these theorems. From the p∨2-power integrability of the process X it follows, by Remark 4.7 in [7], that a weak solution Y = [X, X t ] to (5.3) has square integrable trajectories a.s. Moreover, by Minkowski's integral inequality if X ∈ SL p∨q F (0, t; E), then Y = [X, X t ] ∈ SL p∨q F (0, t; E) (see the proof of Corollary 3.12 in [16]). Finally, notice that from (Hφ) it follows that condition (HA) is satisfied for F = [φ, 0] .
Let α ∈ ( 1 q∨p , 1 2 ) and t > 0. Now we prove that Using the assumption σ ∈ L 2 (0, d), the Cauchy-Schwarz inequality and Fubini's theorem we can estimate the second integral on the right hand side of (5.30) as follows We decompose the first term on the right hand side of (5.30) as Similarly as in the first part of the proof we obtain that σ 2 ∈ L 2 (0, s) for all s > 0. Therefore,