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

Abstract

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: \[ \left(\partial^β+\fracν{2}(-Δ)^{α/2}\right)u(t,x) = I_t^γ\left[ρ(u(t,x))\dot{W}(t,x)\right],\quad t>0,\: x\in\mathbb{R}^d, \] where $\dot{W}$ is the space-time white noise, $α\in(0,2]$, $β\in(0,2)$, $γ\ge 0$ and $ν>0$. 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: $d<2α+\fracαβ\min(2γ-1,0)$. In some cases, the initial data can be measures. When $β\in (0,1]$, we prove the sample path regularity of the solution.

43 pages, 4 figures