The stochastic heat inclusion with fractional time driven by time-space Brownian and Lévy white noise
Olfa Draouil, Rahma Yasmina Moulay Hachemi, Bernt Øksendal
TL;DR
This work analyzes time-fractional stochastic heat inclusions with Caputo derivatives of order $α\in(0,2)$ driven by time-space Brownian and Lévy white noise, incorporating a set-valued drift. It establishes that any solution must be a fixed point of a Volterra-type integral map, and that any fixed point yields a solution, with mild solutions characterized by dimensional and fractional-order constraints. The paper proves existence (via fixed-point arguments) and identifies mild-solution regimes: when $α=1,d=1$ or $α>1$ with $d\in\{1,2\}$, while showing non-mild behavior for all $d$ when $α<1$; an environmental nitrate-transport example illustrates applicability and numerical strategies. The numerical section outlines a finite-difference scheme with memory for the Caputo derivative, random absorption, and additive noise to simulate spatiotemporal evolution, demonstrating the framework’s potential for real-world porous-media problems.
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 $α\in (0,2).$ 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\\ $α=1$ \& $d=1,$ \ or \ $α\geq 1$ \& $d\in \{1,2\}$. On the other hand, if $α< 1$ we show that the solution is not mild for any space dimension $d$.