Space-time fractional diffusions in Gaussian noisy environment

Abstract

This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^β+ \fracν{2} (-Δ)^{α/2} \right) u(t,x)= λu(t,x) \dot{W}(t,x)$, where $\dot W$ is a general Gaussian noise and $β\in (1/2, 2)$, $α\in (0, 2]$. 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.