Space-time fractional stochastic partial differential equations

TL;DR

This paper studies non-linear time-fractional stochastic heat equations in multiple dimensions, proving existence, uniqueness, and continuity of solutions, and explores their relation to higher-order stochastic PDEs, extending prior results to lower dimensions.

Contribution

It extends the theory of stochastic PDEs to include time-fractional derivatives, establishing existence and uniqueness of solutions in dimensions 1 to 3, and connects these equations to higher-order stochastic PDEs.

Findings

Existence and uniqueness of mild solutions in dimensions 1, 2, 3.

Solutions are continuous under certain conditions.

No finite energy solutions for super-linear growth of ta.

Abstract

We consider non-linear time-fractional stochastic heat type equation $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\sigma(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$, $\alpha\in (0,2]$ and $d<\min\{2,\beta^{-1}\}\a$, $\partial^\beta_t$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process, $I^{1-\beta}_t$ is the fractional integral operator, $\stackrel{\cdot}{W}(t,x)$ is space-time white noise, and $\sigma:\RR{R}\to\RR{R}$ is Lipschitz continuous. Time fractional stochastic heat type equations might be used to model phenomenon with random effects with thermal memory. We prove existence and uniqueness of mild solutions to this equation and establish conditions under which the solution is continuous. Our results extend the results in the case of parabolic stochastic partial differential equations obtained in \cite{foondun-khoshnevisan-09, walsh}. In sharp contrast to the stochastic partial differential equations studied earlier in \cite{foondun-khoshnevisan-09, khoshnevisan-cbms, walsh}, in some cases our results give existence of random field solutions in spatial dimensions $d=1,2,3$. Under faster than linear growth of $\sigma$, we show that time fractional stochastic partial differential equation has no finite energy solution. This extends the result of Foondun and Parshad \cite{foondun-parshad} in the case of parabolic stochastic partial differential equations. We also establish a connection of the time fractional stochastic partial differential equations to higher order parabolic stochastic differential equations.