The time fractional stochastic partial differential equations with non-local operator on ${\mathbb{R}}^d$

Abstract

This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on ${\mathbb{R}}^d$ driven by mixed Wiener--Lévy noises. The equations feature a Caputo time derivative ${\partial }_t^{\hat{I}}\pm$ ($0<\hat{I}\pm <1$) and a spatial nonlocal operator $\ddot{I}•(\hat{I}”)$ generated by a subordinate Brownian motion, leading to a doubly nonlocal structure. For the case $p\ge 2$, we prove the existence, uniqueness, and sharp Sobolev regularity of weak solutions in the scale of $\ddot{I}•$-Sobolev spaces ${\mathcal{H}}_p^{\ddot{I}•,{\hat{I}}^3+2}(T)$. Our approach combines harmonic analysis techniques (Fefferman--Stein theorem, Littlewood--Paley theory) with stochastic analysis to handle the combined Wiener and Lévy noise terms. In the special case of cylindrical Wiener noise, a dimensional constraint $d<2\hat{I}{º}_0(2-(2\ddot{I}{ƒ}_2-2/p{)}_{+}/\hat{I}\pm )$ is obtained.~For the low-regularity case $1\le p\le 2$, where maximal function estimates fail, we construct unique local mild solutions in $L_p({\mathbb{R}}^d)$ for equations driven by pure-jump Lévy space-time white noise, using stochastic truncation and fixed-point arguments. The results unify and extend previous theories by simultaneously incorporating time-space nonlocality and jump-type randomness.

39pages