The stochastic heat inclusion with fractional time driven by time-space Brownian and L\'evy white noise

TL;DR

This paper investigates the existence and properties of solutions to a time-fractional stochastic heat inclusion driven by Brownian and Lévy noise, establishing fixed point characterizations and conditions for mild solutions across different fractional orders and dimensions.

Contribution

It introduces a fixed point framework for solutions and determines conditions under which solutions are mild or not, extending understanding of fractional stochastic heat equations with Lévy noise.

Findings

Existence of at least one solution via fixed point theorem.

Mild solutions exist for specific fractional orders and dimensions.

Solutions are not mild when the fractional order is less than one.

Abstract

We study a time-fractional stochastic heat inclusion driven by additive time-space Brownian and L\'evy white noise. The fractional time derivative is interpreted as the Caputo derivative of order $\alpha \in (0,2).$ We show the following: \\ a) If a solution exists, then it is a fixed point of a specific set-valued map.\\ b) Conversely, any fixed point of this map is a solution of the heat inclusion.\\ c) Finally, we show that there is at least one fixed point of this map, thereby proving that there is at least one solution of the time-fractional stochastic heat inclusion. A solution $Y(t,x)$ is called \emph{mild} if $\E[Y^2(t,x)] < \infty$ for all $t,x$. We show that the solution is mild if\\ $\alpha=1$ \& $d=1,$ \ or \ $\alpha \geq 1$ \& $d\in \{1,2\}$. On the other hand, if $\alpha < 1$ we show that the solution is not mild for any space dimension $d$.