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

Abstract

This article is concerned with a forward problem for the following sub-diffusion equation driven by standard Brownian motion \begin{align*} \left( ^{\mathcal C} \partial^\gamma_t + A \right) u(t) = f(t) + B(t) \dot{W}(t), \quad t\in J:=(0,T), \end{align*} where $^{\mathcal C} \partial^\gamma_t$ is the conformable derivative, $\gamma \in (\frac{1}{2},1].$ Under some flexible assumptions on $f,B$ and the initial data, we investigate the existence, regularity, continuity of the solution on two spaces $L^r(J;L^2(\Omega,\dot{H}^\sigma))$ and $C^\alpha(\overline{J};L^2(\Omega,H))$ separately.