The time-fractional stochastic heat equation driven by time-space white noise

TL;DR

This paper investigates the solutions of a time-fractional stochastic heat equation driven by white noise, revealing how the fractional order and spatial dimension influence the existence of mild solutions.

Contribution

It provides explicit solutions for the equation and characterizes the conditions under which solutions are mild based on fractional order and spatial dimension.

Findings

Mild solutions exist for =1 when   (1,2)

Mild solutions exist for =1,2 when   (1,2)

Solutions are not mild for any dimension when <1

Abstract

We study the time-fractional stochastic heat equation driven by time-space white noise with space dimension $d\in\mathbb{N}=\{1,2,...\}$ and the fractional time-derivative is the Caputo derivative of order $\alpha \in (0,2)$. We consider the equation in the sense of distribution, and we find an explicit expression for the $\mathcal{S}'$-valued solution $Y(t,x)$, where $\mathcal{S}'$ is the space of tempered distributions. Following the terminology of Y. Hu \cite{Hu}, we say that the solution is \emph{mild} if $Y(t,x) \in L^2(\mathbb{P})$ for all $t,x$, where $\mathbb{P}$ is the probability law of the underlying time-space Brownian motion. It is well-known that in the classical case with $\alpha = 1$, the solution is mild if and only if the space dimension $d=1$. We prove that if $\alpha \in (1,2)$ the solution is mild if $d=1$ or $d=2$. If $\alpha < 1$ we prove that the solution is not mild for any $d$.