Nonlinear stochastic time-fractional slow and fast diffusion equations on ${\mathbb{R}}^d$

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: $$({\mathrm{\partial }}^{\beta }+\frac{\nu }{2}(-\mathrm{\Delta }{)}^{\alpha \mathrm{/}2})u(t,x)=I_t^{\gamma }[\rho (u(t,x))\dot{W}(t,x)],t>0,x\in {\mathbb{R}}^d,$$ where $\dot{W}$ is the space-time white noise, $\alpha \in (0,2]$, $\beta \in (0,2)$, $\gamma \ge 0$ and $\nu >0$. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: $d<2\alpha +\frac{\alpha }{\beta }\min (2\gamma -1,0)$. In some cases, the initial data can be measures. When $\beta \in (0,1]$, we prove the sample path regularity of the solution.