This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^\beta + \frac{ \nu}{ 2} (-\Delta)^{ \alpha/2} \right) u(t,x)= \lambda u(t,x) \dot{ W}(t,x)$, where $\dot W$ is a general Gaussian noise and $\beta\in (1/2, 2)$, $\alpha\in (0, 2]$. The existence and uniqueness of the solution, the moment bounds of the solution are obtained by using the fundamental solutions of the corresponding deterministic counterpart represented by the Fox H-functions. Along the way, we obtain some new properties of the fundamental solutions.
Categories: math.PR