Well-posedness and regularity of some stochastic time-fractional integral equations in Hilbert space

In the current work, we deal with a class of stochastic time-fractional integral equations in Hilbert space by studying their well-posedness and regularity. Precisely, we use the celebrity fixed point theorem to prove the well-posedness of the problem by imposing the global Lipschitz and the linear growth conditions. Further, we prove the spatial and temporal regularity by imposing only a regularity condition on the initial value. An important example is considered in order to confirm and support the validity of our theoretical results.


Introduction
There are many phenomena in applied sciences as anomalous diffusion in some physical processes and dynamical systems with memory in medicine, which cannot be described adequately by the classical differential and integral equations. This fact gives rise of the theory of fractional differential and integral equations. Such kinds of equations became an effective tool to model such phenomena in the past four decades, although the roots of the fractional calculus that extends to the year 1695, see for a short list [1][2][3][4][5][6][7][8][9].
However, the nondeterministic nature of the most of such phenomena obliges us to incorporate randomness into their mathematical descriptions and provide more realistic mathematical models of them, and this frequently results in stochastic fractional differential and/or integral equations. For example, Arab et al. [10] studied the fractional stochastic Burgers-type equation and proved not only its well-posedness in the Hölder space, but its numerical approximations as well. The authors in Ref. [11] developed the basic theory of fractional calculus and anomalous diffusion from probability's point of view. While Metzler et al. [12] discussed extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.
Recently, a few contributions have been considered by some authors in the analysis of stochastic integrodifferential equations driven by a fractional Brownian motion. For details, Caraballo et al. [13] studied the asymptotic behaviour for neutral stochastic integrodifferential equations driven by a fractional Brownian motion. Arthi et al. [14] proved the existence and exponential stability of a same class of equations with impulses. In Ref. [15], Sathiyara et al. dealt with the contrabillity of fractional higher order stochastic integrodifferential systems with fractional Brownian motion. Moreover, a class of fractional stochastic Itö integral or Skorokhod integral equations has been studied by El-borai et al. [16,17], where the well-posedness of such classes has been established and in the recent literature various qualitative behaviors of solutions of Volterra integral equations, ordinary and stochastic differential delay equations of second order and integrodifferential equations of first order have been investigated in [18][19][20][21][22] and some new qualitative results have been obtained in these papers.
From these facts and regarding the importance of the stochastic time-fractional integral equations in the description of some anomalous diffusions, our contribution in the current paper will be the study of such class of equations, which is given in the following general form: where, t ∈ [0, T], for fixed T > 0, α ∈ ( 1 2 , 1], the initial condition u 0 := u(0) is a H-valued F 0 -measurable random variable, with H be a real Hilbert space, A : D(A) ⊂ H → H is a linear closed operator generates a semigroup (S(t)) t∈ [0,T] , F : H → H and G : H → H are two operators, W is a H-valued cylindrical Wiener process.
In our work, we consider the fractional integrals in the Riemann-Liouville approach.
It is worth mentioning that in Ref. [16], the authors studied an equation similar to (1), perturbed by a multiplicative noise, where W is a Wiener process on the real separable Hilbert space K, with covariance operator Q. They proved only the well-posedness of the problem, where the mild solution (u(t)) t∈[0,T] satisfies u(t) is an L 2 ( , H)-valued random variable, for any t ∈ [0, T]. In our case, the equation is perturbed by an additive noise, where we prove the well-posedness, with u(t) is an L p ( , H)-valued random variable, for p ≥ 2. Further, to the best of the authors knowledge, there is no work has been dealt with the regularity of solutions for such class of equations. For this, our main contribution is not only to study the temporal regularity, but to study the spatial regularity as well.
This work is ordered by the following: Section 2 is devoted to prove the well-posedness of the problem. In Section 3 we prove the spatial regularity of the problem. The temporal regularity is postponted to Section 4. Finally, conclusion is presented in Section 6.
We close this introduction by giving the following notations. For O an operator we mean by D(O) its domain of definition, H is a real Hilbert space endowed with the norm |.| H , L(H) is the space of linear bounded operators defined on H into it self endowed with the norm . L(H) , HS is the space of Hilbert-Schmidt operators defined from the Hilbert space H into it self, and we indicate its norm by . HS . Let ( , F, F, P) be a filtered probability space, where F := (F t ) t∈[0,T] is a normal filtration and X be a Banach space, L p ( , X), for p ≥ 2 is the space of X-valued pth integrable random variables on , its norm is denoted by . L p ( ,X) , H) . We use the abbreviation RHS for right hand side.

Well-posedness of problem (1)
The current section is devoted to prove the existence and the uniqueness of a mild solution u defined as follows (see Refs. [16,23,24]).
• u satisfies the following equality in H, P-a.s., where ξ α is a probability density function defined on (0, ∞).
We need here to impose the following assumptions to prove the well-posedness of problem (1). For p ≥ 2: and and for some positive constant C F . We need here to reformulate Assumption H F in the random context. To this end, let x and y be two H-valued random variables. Then, we have and H u 0 -The initial condition u 0 is an F 0 -measurable random variable, satisfies u 0 ∈ L p ( , F 0 , P; H), i.e. u 0 L p ( ,H) < ∞.

Remark 2.1:
In the rest of this paper, when we need to use estimations in the random context as it has been proved above for Assumption H F , we will do it without proof in order to avoid the repetitions.

Theorem 2.1: Under the Assumptions H
To prove this theorem, we need the following useful result.

Proof of Theorem 2.1:
Let α ∈ ( 1 2 , 1) and p ≥ 2. We define the mapping ψ : First we have, To estimate To estimate the second term in the RHS of (10), let α ∈ ( 1 2 , 1). Then, by using Assumption Now, to deal with the stochastic term in the RHS of (10), we use Burkholder-Davis-Gundy inequality (see [25, Proposition 2.12, p.24]) as follows HS . To do this, let α ∈ ( 1 2 , 1) and γ ∈ (0, 1 − 1 2α ), by using the fact that AB HS ≤ A HS B L(H) , for every A ∈ HS and every B ∈ L(H), we obtain And so, Thanks to the condition γ ∈ (0, 1 − 1 2α ), we can easily obtain Coming back to Est.(10), we replace Est. (11), Est. (12) and Est. (13) in it, to get For all t ∈ [0, T]. And so is well defined. Now, we prove that is a contraction mapping. To do this, let u, v ∈ ([0, T]; H). From Est. (9) we infer that To estimate the term in the RHS of Est. (15), we follow the same steps to lead to Est. (12). Thus Consequently, If C S C α,1 C F T α < 1, then ψ is a contraction, and so it has a unique fixed point that coincides with a unique mild solution of problem (1).

Spatial regularity of problem (1)
In this section, we study the spatial regularity of the mild solution. Its main result is the following.
for some two positive constants C 1 and C 2 . Then, there exists C = C (C 2 ,T, ) > 0, such that

Temporal regularity of problem (1)
We deal with the temporal regularity of the mild solution in this section, where its main result is the following.

Theorem 4.1: According to Assumptions H A , H F and
H G , the mild solution u of (1) with an initial condition u 0 satisfies A η u 0 L p ( ,H) < ∞, for all η ∈ (0, 1 − 1 2α ), is time Hölder continuous with exponent μ := min{αη, for all 0 < t 0 ≤ s < t ≤ T, with t 0 and T be fixed and for some positive constant C.
In order to prove Theorem 4.1, we need the following useful lemma besides Lemma 2.1.

Example: stochastic space-time fractional integro-differential equation
We give in this section an important example of problem (1) in order to confirm and support the validity of our theoretical results. Namely; Theorems 2.1, 3.1 and 4.1.

Conclusion
Stochastic fractional integral (or integro-differential) equations have been used as mathematical models of many physical processes as the anomalous diffusions. Nevertheless, we can find in the literature a few works concerned with this type of equations. Motivated by this fact, we considered in this paper a class of stochastic time-fractional integral equations in a Hilbert space H. We used the fixed point theorem in order to prove the well-posedness of the problem by imposing the global Lipschitz and the linear growth conditions. Moreover, we achieved not only the spatial regularity of the mild solution u := (u(t)) t∈[0,T] but the temporal regularity as well. Precisely, by imposing a regularity condition on the initial value (i.e. A η u 0 L p ( ,H) < ∞), we proved that such solution satisfies u ∈ C([0, T]; L p ( , D(A η ))), where η ∈ (0, 1 2 − 1 4α ), p ≥ 2 and it is time Hölder continous with exponent μ := min{αη, 1 − α, α(1 − η) − 1 2 }, for α ∈ ( 1 2 , 1) and η ∈ (0, 1 − 1 2α ). In general, it is not easy to solve this kind of equations analytically, for this, the numerical study plays an important role by providing a numerical approximations of the analytic solutions with respect to time, space or to both simultaneously. Motivated by this fact, some numerical approximations for the mild solution of the problem are interesting directions for our future research.