The stochastic heat inclusion with fractional time driven by time-space Brownian and Lévy white noise

Abstract

We study a time-fractional stochastic heat inclusion driven by additive time-space Brownian and Lévy 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 \ $\alpha \ge 1$ \& $d\in \{1,2\}$. On the other hand, if $\alpha <1$ we show that the solution is not mild for any space dimension $d$.