Hölder Continuous Regularity of Stochastic Convolutions with Distributed Delay

In this work, we consider the Hölder continuous regularity of stochastic convolutions for a class of linear stochastic retarded functional differential equations with distributed delay in Hilbert spaces. By focusing on distributed delays, we first establish some more delicate estimates for fundamental solutions than those given in Liu (Discrete Contin. Dyn. Syst. Ser. B 25(4), 1279–1298, 2020). Then we apply these estimates to stochastic convolutions incurred by distributed delay to study their regularity property. Last, we present some easily-verified results by considering the regularity of a class of systems whose delay operators have the same order derivatives as those in instantaneous ones.


Introduction
To motivate this work, let a : [−r, 0] → R, r > 0, be some appropriate function and h 0 : R → R be a monotonically continuous function which satisfies the following relations (see, e.g., Coleman and Gurtin [1] and Nunziato [7]) where p ≥ 2, γ i > 0 and α i ∈ R for i = 1, 2. The non-Fourier heat conduction model with memory in the conductor (0, π) ⊂ R may start from the constitutive equation x ∈ (0, π), (1.1) and the energy conservative equation without exterior energy sources d t p(t, x) + ∂q(t, x) ∂x dt = 0, t ≥ 0, x ∈ (0, π), (1.2) where y denotes the temperature, q is the heat flux and p is the internal energy which can be taken, in most situations, as the form: p(t, x) = κy(t, x), κ > 0. In practice, the assumption of zero exterior energy source is artificial, and a more realistic model is that the null exterior energy source is perturbed by a noise, for example, a white noise random field b(x)ẇ(t, x), b ∈ L 2 (0, π). In other words, Eq. 1.2 is replaced at present by the following equation Then, by substituting Eq. 1.3 into 1.1, one can obtain, if κ = 1, an equation of the form y(0, ·) = φ 0 (·) ∈ L 2 (0, π), y(θ, ·) = φ 1 (θ, ·) ∈ W 1,p 0 (0, π), θ ∈ [−r, 0], y(t, 0) = y(t, π) = 0, t ∈ (0, ∞). (1.6) The aim of this work is to investigate the regularity property of such stochastic systems as Eq. 1.6. The organization of this work is as follows. In Section 2, we first introduce a deterministic linear retarded functional differential equation associated in our formulation of stochastic systems. We review the useful variation of constants formula for the equations under consideration by means of a method of fundamental solutions. Also, we show some estimates of fundamental solutions which will play an important role in the subsequent development. In Sections 3 and 4, we give the detailed proofs of the key result, i.e., Theorem 2.1, by following Prüss's method of constructing some resolvent operators for the integrodifferential equations of Volterra type. By employing the main results, we establish in Section 5 the desired regularity property for stochastic convolutions. Last, we focus on some particular cases such as A = A 1 in Eq. 5.2, which allow us in Section 6 to deal with a class of systems with delays in the highest-order derivatives and in Section 7 to consider the continuity of stochastic convolutions not in a Hilbert space, but in a smaller Banach space of this space.
Finally, a few words about the notation are in order: generic positive constants will be denoted by C; we shall use the shorthand notation a b to denote a ≤ Cb. If the constant C depends on a parameter p, we may write C p and a p b.

Fundamental Solution
Let H be a separable Hilbert space equipped with the norm · H and inner product ·, · H , respectively. Also, we denote by L (H ) the space of all bounded linear operators in H and by · the usual operator norm in L (H ). We are concerned with the following linear retarded functional differential equation in H , where r > 0 is some constant incurring the system delay, a(·) ∈ L 2 ([−r, 0], R) and φ = (φ 0 , φ 1 ) is an appropriate initial datum. Here A : D(A) ⊂ H → H is the infinitesimal generator of an analytic semigroup e tA , t ≥ 0, A 1 is a closed linear operator with domain D(A 1 ) ⊃ D(A) and f is some continuous function with values in H . For simplicity, we assume in this work that the C 0 -semigroup e tA is negative type, i.e., there exist constants M ≥ 1 and μ > 0 such that and for γ ∈ (0, 1), there exists a constant M γ > 0 such that where (−A) γ is the standard fractional power of operator A. Equations of the type (2.1) were investigated by many researchers such as Di Blasio, Kunisch and Sinestrari [3] and the fundamental solution to Eq. 2.1 was introduced by Jeong, Nakagiri and Tanabe [5]. In particular, it is known that the fundamental solution G(·) : R → L (H ) to Eq. 2.1 is an operator-valued function which is strongly continuous in H and satisfies the integral equation where O is the null operator in H . The fundamental solution G enables us to solve the initial value problem (2.1). In fact, it may be shown that under some reasonable conditions on f and initial datum φ = (φ 0 , φ 1 ), the unique mild solution y to Eq. 2.1 is represented as with the initial condition y(0) = φ 0 and y(θ) = φ 1 (θ), θ ∈ [−r, 0). This is a time delay version of the usual variation of constants formula without memory.
In order to apply Eq. 2.5 to such equations as Eq. 1.6 to consider their regularity property of solutions, we need establish some inequalities in association with G(·). To this end, we introduce the following conditions: is Hölder continuous, i.e., there exist constants M > 0 and ρ ∈ (0, 1) such that and A 1 e tA = e tA A 1 for each t > 0 or both A and A 1 are self-adjoint.
In other words, Theorem 2.1 here improves the results (2.11), (2.12) in [6] for systems with distributed delay.

Remark 2.2
In this work, we are mainly interested in unbounded operator A 1 . If A 1 is bounded, Eq. 2.6 is clearly valid for ν = 0 and vise versa, and then we actually come to a trivial case. In this situation, all the results in Theorem 2.1 remain true for γ ∈ (0, 1) and their proofs can be significantly simplified in comparison with those in the following sections.

Proposition 3.1 The mapping (·) : [0, r] → L (H ) is uniformly bounded and for any
To show Theorem 2.1 (a), we intend to develop an induction scheme. We first consider the case that n = 0 and set Then, the integral equation to be satisfied by V (t) is is defined on all of H and by the well-known closed graph theorem we have A 1 A −1 ∈ L (H ). Then for any 0 < β < 1 − γ , it follows by virtue of Proposition 3.1 and Eq. 2.2 that Hence, by virtue of the well-known Gronwall lemma, Eqs. 3.4 and 3.5, it is easy to have

Now assume that condition (H2) holds, then we have by the relation
Also, it is easy to have that for any u ∈ D(A) and t > 0, which, by virtue of Theorem 2.3.4, pp. 45-46, [9], immediately implies that for any x ∈ H , Hence, for t ∈ (0, r] we have which is Eq. 2.8 with n = 1. Now suppose the fundamental solution G(·) satisfies all the estimates in Theorem 2.1 (a) on the intervals [0, nr]. Then in the interval (nr, (n + 1)r], the integral equation to be satisfied by is If condition (H1) holds, it is known by Liu [6] that Eq. 2.8 is true on [0, (n + 1)r]. Now assume that (H2) holds and we estimate each term on the right hand side of Eq. 3.11. First, note that It is known by Theorem 1 in Tanabe [8] that for i = 0, 1, A 0 = A, for some C j > 0, j ∈ {0, 1, 2, · · · }. To estimate I 2 (t), we first note that for any 0 < s < t < ∞ and α ∈ (0, 1] (cf. (4.23) in [6]). Then we re-write I 2 (t) as For j = 0, 1, · · · , n − 2 and t ∈ (nr, (n + 1)r], we have by Eqs. 3.13 and 3.14 that where B(·, ·) is the standard Beta function. In a similar manner, we have for t ∈ (nr, (n + 1)r] that By virtue of Eqs. 3.15, 3.16 and 3.17, we thus obtain for t ∈ (nr, (n + 1)r] that On the other hand, we have by using Eqs. 3.3 and 3.13 that where 0 < β < 1 − γ and C β,γ , C n−1 > 0. Combining Eqs. 3.12, 3.18 and 3.19, we thus have

Proof of Theorem 2.1 (b)
We first consider V 0 (t) and Proof If condition (H1) holds, it is known by Liu [6] that Eq. 4.1 is true. Now assume that (H2) holds and we have, by definition, that for 0 < s < t ≤ r, Note that log(1 + a) ≤ a α /α for any a > 0, 0 < α ≤ 1. Therefore, for 0 < s < t ≤ r, it is easy to see from Eq. 4.3 that and in a similar manner, by Proposition 3.1, we have for 0 < s < t ≤ r that To estimate J 2 (·.·), note that for 0 < u < s < t ≤ r and some δ > 0 sufficiently small, we have by using Proposition 3.1 that (4.6) and Hence, we have from Eqs. 4.6 and 4.7 that which, in addition to Eq. 4.2, implies Hence, by combining Eqs. 4.4, 4.5 and 4.9, we get for 0 < s < t ≤ r that from which the desired result follows by the well-known Gronwall inequality and Eq. 3.4.
(I) We first assume that condition (H1) holds. It is noticed that on the intervals [nr, (n + 1)r], n ≥ 1, the integral equation to be satisfied by is and V 0 (t) is given as in Eq. 3.11. We first show the Hölder continuity of V 0 (t) and V (t).
Let nr < s < t ≤ (n + 1)r. By virtue of Eq. 3.11, we have First, by virtue of Eq. 4.10 we have As for the second term on the right side of Eq. 4.12, we have

e (t−r−u)A − e (s−r−u)A − e (t−nr)A + e (s−nr)A ≤ e (t−r−u)A − e (s−r−u)A + e (t−nr)A − e (s−nr)
In a similar way, we have by Eq. 3.14 that

Now we intend to consider I 3 (t, s).
For sufficiently small δ > 0 and 0 < β < 1 − γ , we may obtain by virtue of Proposition 3.1 that Combining Eqs. 4.12-4.21, we thus conclude that for nr < s < t ≤ (n + 1)r, and further we have (II) Now assume that condition (H2) holds. Note that on the intervals [nr, (n + 1)r], n ≥ 1, the integral equation to be satisfied by t ∈ [nr, (n + 1)r], (4.23) is and V 0 (t), given in Eq. 3.11, can be re-written as =: I 1 (t, s) + I 2 (t, s) + I 3 (t, s) + I 4 (t, s) + I 5 (t, s) + I 6 (t, s), nr < s < t ≤ (n + 1)r. (4.24) First, by virtue of Eq. 4.10 we have It is known by Proposition 4.2 in Jeong [4] that for i = 0, 1, A 0 = A, To handle I 3 (·, ·), we have by virtue of Eq. 3.14 that for 0 < u < s − r and sufficiently small δ > 0, (4.28) In a similar way, we have by Eq. 3.14 that for 0 < β < 1 − γ and sufficiently small δ > 0. By using Eqs. 4.26 and 4.30, we have for sufficiently small δ > 0 that On the other hand, we have by using Eqs. 3.14 and 4.26 that For I 5 (t, s), we may obtain by virtue of Proposition 3.1 and Eq. 4.26 that (4.33) Last, for I 6 (t, s) and nr < s < t ≤ (n + 1)r, we have by virtue of Eqs. 4.3 and 4.26 that Combining Eqs. 4.24-4.34, we thus conclude that and further for nr < s < t ≤ (n + 1)r. Finally, by combining Eqs. 4.22 and 4.35, we have for any (n − 1)r < s ≤ nr < t ≤ (n + 1)r, n > 1, that Hence, Eq. 4.22 holds for r < s < t ≤ nr and t − s < r, n > 1 with constants independent on s and t, and further we obtain the desired estimates Eq. 2.10. The proof is thus complete.

Stochastic Convolution
Let { , F , P} be a probability space equipped with some filtration {F t } t≥0 . Let K be a separable Hilbert space and {W Q (t), t ≥ 0} denote a Q-Wiener process with respect to {F t } t≥0 in K, defined on { , F , P} where Q is a positive, self-adjoint and trace class operator on K. We frequently call W Q (t), t ≥ 0, a K-valued Q-Wiener process with respect to {F t } t≥0 if the trace T r(Q) < ∞. We introduce a subspace K Q = R(Q 1/2 ) ⊂ K, the range of Q 1/2 , which is a Hilbert space endowed with the inner product Let L 2 (K Q , H ) be the space of all Hilbert-Schmidt operators from K Q into H . Suppose that W (·) is a Q-Wiener process in K such that Qe j = λ j e j , j ≥ 1, λ j > 0, (5.1) where {e j } is a complete orthonormal basis in K, then it is immediate that where {w j (·)} is a group of independent real Wiener processes.
We are concerned about the following linear stochastic retarded functional differential where A, A 1 are given as in Section 2, B ∈ L 2 (K Q , H ) and φ = (φ 0 , φ 1 ) is an appropriate initial datum. It is well known that the unique mild solution y of Eq. 5.2 is represented as with the initial condition y(0) = φ 0 and y(θ) = φ 1 (θ), θ ∈ [−r, 0). In particular, if φ = (0, 0), then the unique mild solution Eq. 5.3 is the so-called stochastic convolution process For simplicity, let us assume at present that K = H and B = I in Eq. 5.2. Proof We intend to use a Kolmogorov test type of argument (see, e.g., Theorem 3.5 in Da Prato and Zabczyk [2]). For any 0 ≤ s < t ≤ T , by definition, it is easy to see that Now we estimate I 1 and I 2 , respectively. Since 0 < β < 1 2 − γ , it follows that 1 − 2γ > 2β > 0. By using Theorem 2.1 (a), we thus have Note that 1 − 2β − 2γ > 0. If s ≤ r, we employ Theorem 2.1 (b) to obtain If s > r, we similarly employ Theorem 2.1 (b) to obtain Hence, by substituting Eqs. 5.6, 5.7 and 5.8 into 5.5, we obtain is Gaussian, for any integer m ≥ 1, it follows by Eq. 5.9 that So, by the well-known Kolmogorov test, (−A) γ W G (·) is α m -Hölder continuous with order Since m is arbitrary, the trajectories of process (−A) γ W G are thus in C β ([0, T ]; H ).

Delays in the Highest-Order Derivatives
In the remainder of this work, we assume that the operators A and A 1 in Eq. 5.2 are diagonal with respect to the complete orthonormal basis {e k } given in Eq. 5.1 such that for some numbers μ k > 0,μ k ∈ R, k ∈ N = {1, 2, · · · }. For α, β ∈ R, let g(t), t ≥ 0, denote the unique solution of the following differential equation with delay and the fundamental solution G is determined by

3)
Proof First note that for any k ∈ N, 0 ≤ s < t ≤ nr and v > 0, where {e k } is the orthonormal basis of H in Eq. 6.1. Hence, by virtue of Eq. 2.9 in Theorem 2.1 (b) we have for any 0 ≤ s < t ≤ nr and 0 < v ≤ r that (6.10) or similarly, for any 0 ≤ s < t ≤ nr and v > r that If s ∈ [0, r], it is easy to see from Eq. 6.10 that On the other hand, if s > r, one can similarly have from Eqs. 6.10 and 6.11 that where M γ,β,n = 1 1 − β C 2 n,β,γ · [(n − 1)r] 1−β + The proof is thus complete.
Proof It suffices to prove this theorem for any 0 ≤ s < t ≤ Nr and fixed N ∈ N. First, by virtue of Eqs. 6.5 and 7.3 we have that for ε > 0 small enough with δ − ε > 0, If s > r, then we have by Eqs. 7.9 and 7.10 that Combining Eqs. 7.8 and 7.12, we have the conclusion that there exists C δ,ε > 0 such that E|W G (t, ξ ) − W G (s, ξ )| 2 ≤ C δ,ε (t − s) δ−ε , for any 0 ≤ s < t ≤ Nr.
Further, for any 0 ≤ s < t ≤ Nr, there exists some C δ,ε > 0 such that Since W G (t, ξ ) − W G (s, η) is a Gaussian random variable, for any m ∈ N we thus have for some number c m,δ,ε > 0 that .
Choosing m such that m(δ − ε) > 1 and applying the well-known Kolmogorov test to random fields, we can find that W G (·, ·) is α-Hölder continuous for Let m → ∞ and ε → 0, then we have the desired result.