Stochastic sub-di usion equation with conformable derivative driven by standard Brownian motion

This article is concerned with a forward problem for the following sub-di usion equation driven by standard Brownian motion (C∂γ t +A)u(t) = f(t) +B(t)Ẇ (t), t ∈ J := (0, T ), where C∂ t is the conformable derivative, γ ∈ (12 , 1]. Under some exible assumptions on f,B and the initial data, we investigate the existence, regularity, continuity of the solution on two spaces Lr(J ;L2(Ω, Ḣσ)) and Cα(J ;L2(Ω, H)) separately.


Introduction
The normal diusion processes are characterized by a linear growth in a time of the mean squared displacement (variance increases linearly in time). Besides, there is also a concept of anomalous diusion regimes. They are characterized by a variance growing slower (sub-diusion) or faster than normal (superdiusion). Many dierent underlying processes can lead to anomalous diusion, with qualitative dierences between mechanisms producing sub-diusion and mechanisms resulting in super-diusion.
Recent decades have witnessed the escalating popularity of fractional calculus, which replaces the classical time derivative in a partial dierential equation with a fractional derivative. Because of containing information of the considered function at the previous time, fractional derivatives in time (for instance, Caputo, RiemannLiouville, and Conformable) reect memory, history, and non-local spatial eects, which are paramount crucial in better modeling and understanding of the complex and dynamic system behavior. We can list here some successful applications of fractional calculus to specic problems of dynamics [18,38,43]. Some applications of Conformable derivative in modeling neuronal dynamics, dynamic cobweb, dynamics of a Particle in a viscoelastic medium, and fractional-order chaotic system can be found in [15,16,24,32,55]. One of the ecient and widely used approaches to modeling sub-diusion is based on the theory of fractional derivatives. Such equations were widely used to model many phenomena in nature. For example: Population dynamics of water conned in soft environments [33]; Transport phenomena of proteins in cellular environments [10,27,35,45]; Ion movement in dendritic spines [28,50], and RNA molecules within cells [31]. Some interesting works can be found in [2,3,4,5,6,7,8,9,63,64,65].
Let T be a positive constant, γ ∈ ( 1 2 , 1] and J := (0, T ). Let D be a bounded subset of R d , with d ≥ 1, and possess a smooth boundary when d > 1. Let H := L 2 (D) and A = −∆ be the negative Laplacian. Without loss of generality, the operator A is assumed to possess eigenpairs (λ k , e k ) satisfying 0 < λ k ∞ and Ae k = λ k e k , for every k ≥ 1.
Let {W (t)} t∈J be a standard Brownian motion (sBm) taking value in H (see Section 2 for more details). This article is concerned with the following problem for a stochastic conformable diusion equation (SCDE) perturbed by sBm whereẆ (t) = ∂W (t)/∂t describes a white noise, ϕ : Ω → H is known as the initial value, f : J × Ω → H is called the source function, and B is an operator coming from J to L 2 0 dened in Section 2. The notation C ∂ γ t , with γ ∈ ( 1 2 , 1], stands for the conformable derivative [36] dened as which was invented by Khalil in [36] (refer to [1,12,29,59] for some other works). Noting that if γ = 1 and there is no the appearance of the stochastic term B(t)Ẇ (t), then the SCDE in (1) becomes the normal deterministic diusion equation, which was considered in [20,21]. With the appearance of the conformable derivative, an alternative expression of the diusion equation (called conformable diusion equation) is proposed to improve the modeling of anomalous diusion (see [59]). This new fractional derivative has been found to possess many successful applications of in many elds of science [11,23,32,42,44,53,54]. In recent time, the number of articles concerning with the deterministic conformable diusion equation has increased signicantly [1,13,14,29,39,54,56]. For some recent studies on diusion equations with random noise, the readers can refer to [51,52,62].
Stochastic partial dierential equations (SPDEs) are crucial issues modeling the phenomena in a lot of elds of science [19,34,46,48]. Additionally, it is eective to use fractional dierential equations (FDEs) to model some anomalous diusion phenomena in physics, chemistry, engineering, etc. [22, 37, 47, 49, 57, 58]. The area of SFDEs is interesting to mathematicians since it contains various hard open problems [30,40,41,61]. It is a fact that our considered equation in this paper is included in the topic of SFDEs. Notwithstanding the importance, as we know, there is no result concerning the initial value problem (or called forward problem) for SCDE (1). This motivated us to contribute the existence, regularity, and continuity results for Problem (1).
We now mention the organization of this paper. Section (2) introduces notations, functional spaces, and the denition of the mild solution. Section 3 is divided into two subsections. Subsection 3.1 gives some prior estimates for the terms appearing in the representation of the solution. In Subsection 3.2, the existence, regularity, and continuity results are stated.

Preliminaries
For ς > 0, we dene byḢ ς the following spacė and denote byḢ −ς its dual space. We dene by A σ :Ḣ ς/2 →Ḣ −ς/2 [17, 26] the following operator Let Q be a covariance operator [25,60] on H with nite trace, i.e. T r(Q) = k≥1 χ k < ∞, and satisfy Qe k = χ k e k , where {χ k } k≥1 is the spectrum of Q. Assume that {Ẇ (t)} t∈J be an H-valued Wiener process dened on a complete probability space (Ω, F, P, {F t } t≥0 ), with covariance operator Q and posses the following representation (see [25,60]) where ξ k (t) are independent one-dimensional Brownian motions. Let L 2 (Ω,Ḣ ς ) be the Bochner space dened by Let L(H,Ḣ ς ) be the space of bounded linear operators T coming from H toḢ ς . Let HS(H,Ḣ ς ) be the space of all operators T ∈ L(H,Ḣ ς ) such that T HS(H,Ḣ ς ) := k≥1 Te k 2Ḣ ς 1/2 < ∞. By L 2 0 (H,Ḣ ς ) we denote the space of T ∈ HS(H,Ḣ ς ) such that In the case ς = 0, it is obvious thatḢ 0 = H. For short, we denote L(H) := L(H, H), L 2 0 := L 2 0 (H, H). Given a Hilbert space K. For p ≥ 1, we dene the following space Let C(J; K) be space of continuous functions ψ : J → K such that For α > 0, we recall the following subspace of C(J; K) Now, we aim nd an expression for u(t) in the form u(t) = k≥1 (u(t), e k )e k . From the SCDE in (1), we immediately obtain By using the method in Theorem 5 of [29] and Theorem 3.3 of [39], we arrive at Using u(0) = ϕ, we obtain the following equation Basing the above expression for u(t), we dene mild solution.
1. An H-valued process {u(t)} t∈J is said to be a mild solution of (1) if for almost all t ∈ J, it satises the equation (2) almost surely.
Let σ be a non-negative constant satisfying σ < 1 2 . The following assumptions are needed to establish our results.

Some prior estimates
This subsection is aimed to give some prior estimates which will be used throughout this paper. From now on, we employ the notation a 1 a 2 to describe a 1 ≤ Ca 2 , where C is a positive constant.

Proof. i) By applying the inequality
Now, we aim to show the following estimate holds for every k ≥ 1 The Hölder inequality associated with the property (5) allow that where it should be noted that 1 − 2σ > 0 since σ ∈ (0, 1 2 ). Next, we will use the property (6) to prove that (3) holds. Indeed, we have In addition, by using the Hölder inequality again, we obtain Combining (7), (8), we now obtain the desired estimate (3). ii) Firstly, for δ > 0, we can see that The rst term can be estimated by using the inequality By the Hölder inequality and noting that e Combining (10), (11), we obtain Next, we continue to estimate Z 2 (t, δ). We rst see that In addition, the Hölder inequality and e From (13) and (14) and noting that k≥1 Using (t + δ) γ − t γ ≤ δ γ and a similar estimate as in (8), we arrive at Now, combining (9), (12), (15), we conclude that the estimate (4) holds.
Proof. i) By a similar way as in (5), we also have e This together with the denition of the norm in L 2 0 yields By the Itô isometry and the above estimate, it is obvious that The Hölder inequality allows that By using the substituting method to calculate the rst integral, we arrive at where β(., .) is the beta function. It should be noted that c 1 , c 2 > 0, since q > max( 2 2γ−1 , 2 1−2σ ). Now, from three later estimates, we obtain (16) as desired.
ii) Let us turn our attention to estimate two terms dened as follows We begin to estimate the rst term I 1 (t, δ) by using the Itô isometry as Consider the expectation under the integral sign, we can see The By using the above property, we can deduce that Combining (22) and (23), we obtain From (21) and (24), we deduce that By the same way as in (18)-(19), one can check that where c 1 = 1 − (1−γ)(q+2) γ(q−2) , c 3 = 1 − 2(1 − s) q q−2 . In the above estimate, it should be noted that c 1 , c 3 > 0 and c 1 + c 3 − 1 > 0, since q > max( 2 2γ−1 , 2 2s−1 ). Next, we continue to estimate the second term I 2 (t, δ) by using the Itô isometry as This together with the Hölder inequality gives us Now, from (20), (25) and (26), we obtain (17) as desired.
Now, we are ready to state our main results.

The existence, regularity and continuity
Firstly, let us state the following theorem which shows an existence, uniqueness, regularity result on L r (J; L 2 (Ω,Ḣ σ )).
Proof of Corollary 3.1. This result can be proved easily by applying Theorem 3.1. Here, it should be noted that, in the case of σ = 0, the conditions in (27) always hold.
Next, we state another main result considered in the space C α (J; L 2 (Ω, H)).
Proof. It is clear to see that H) .

Using the fact that
which leads to the continuity results (36)

Conclusion
In the present article, a forward problem (or called initial value problem) for a sub-diusion equation perturbed by standard Brownian motion is investigated. With the support of stochastic analysis, we obtain some sucient conditions ensuring the existence, regularity and continuity of the mild solution of such problem.