Asymptotic properties of some space-time fractional stochastic equations

TL;DR

This paper investigates the long-term behavior of solutions to non-linear space-time fractional stochastic heat equations involving fractional derivatives and stable processes, extending previous results in the field.

Contribution

It provides new asymptotic analysis of solutions to complex fractional stochastic equations, including properties of their deterministic counterparts.

Findings

Asymptotic behavior characterized for large time and parameter values

Extended existing results on fractional stochastic equations

Derived properties of the deterministic equations

Abstract

Consider non-linear time-fractional stochastic heat type equations of the following type, $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\lambda \sigma(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$, $\alpha\in (0,2]$. The operator $\partial^\beta_t$ is the Caputo fractional derivative while $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process and $I^{1-\beta}_t$ is the fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. And the multiplicative non-linearity $\sigma$ is assumed to be globally Lipschitz continuous. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter $\lambda$. In particular, our results are significant extensions of existing results. Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.