We study the stochastic time-fractional stochastic heat equation $$\begin{aligned} \frac{\partial ^{\alpha }}{\partial t^{\alpha }}Y(t,x)=\lambda \varDelta Y(t,x)+\sigma W(t,x);\; (t,x)\in (0,\infty )\times \mathbb {R}^{d}, \end{aligned}$$ ∂ α ∂ t α Y ( t , x ) = λ Δ Y ( t , x ) + σ W ( t , x ) ; ( t , x ) ∈ ( 0 , ∞ ) × R d , where $$d\in \mathbb {N}=\{1,2,...\}$$ d ∈ N = { 1 , 2 , . . . } and $$\frac{\partial ^{\alpha }}{\partial t^{\alpha }}$$ ∂ α ∂ t α is the Caputo derivative of order $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) , and $$\lambda >0$$ λ > 0 and $$\sigma \in \mathbb {R}$$ σ ∈ R are given constants. Here $$\varDelta $$ Δ denotes the Laplacian operator, W(t, x) is time-space white noise, defined by $$\begin{aligned} W(t,x)=\frac{\partial }{\partial t}\frac{\partial ^{d}B(t,x)}{\partial x_{1}...\partial x_{d}}, \end{aligned}$$ W ( t , x ) = ∂ ∂ t ∂ d B ( t , x ) ∂ x 1 . . . ∂ x d , $$B(t,x)=B(t,x,\omega ); t\ge 0, x \in \mathbb {R}^d, \omega \in \varOmega $$ B ( t , x ) = B ( t , x , ω ) ; t ≥ 0 , x ∈ R d , ω ∈ Ω being time-space Brownian motion with probability law $$\mathbb {P}$$ P . We consider the equation (0.1) in the sense of distribution, and we find an explicit expression for the $$\mathcal {S}'$$ S ′ -valued solution Y(t, x), where $$\mathcal {S}'$$ S ′ is the space of tempered distributions. Following the terminology of Y. Hu [11], we say that the solution is mild if $$Y(t,x) \in L^2(\mathbb {P})$$ Y ( t , x ) ∈ L 2 ( P ) for all t, x. It is well-known that in the classical case with $$\alpha = 1$$ α = 1 , the solution is mild if and only if the space dimension $$d=1$$ d = 1 . We prove that if $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) the solution is mild if $$d=1$$ d = 1 or $$d=2$$ d = 2 . If $$\alpha < 1$$ α < 1 we prove that the solution is not mild for any d.
In this paper, we study the existence of mild solutions of Hilfer fractional stochastic differential equation with impulses driven by sub-fractional Brownian motion. The results are obtained by using Burton-Kirk's fixed point theorem. In the end, an example is given to illustrate the obtained results.
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