Abstract:

In this paper, we will study a class of fractional stochastic heat equation of the form $\frac{ \partial}{ \partial t} u(t, x)={\cal D}_\delta^\alpha u(t, x)+g( u(t, x)) \frac{\partial^2}{\partial t\partial x}w_\rho(t, x), \quad 0<t\leq T, \quad x\in{\Bbb R} ,$ with $T>0$, where ${\cal D}_\delta^\alpha$ denotes a nonlocal fractional differential operator with $\alpha\in(1, 2]$ and $|\delta|\leq2-\alpha$, and $\frac{\partial^2}{\partial t\partial x}w_\rho(t, x)$ is a spatially inhomogeneous white noise. Under some mild assumptions on the catalytic measure of the inhomogeneous Brownian sheet $w_\rho(t, x)$, we prove the existence, uniqueness and Hölder regularity of the solution. Upper and lower moment bounds for the solution are also derived.

Key words: Fractional stochastic heat equation, Spatially inhomogeneous white noise, Hölder regularity, Moment bounds