The time fractional stochastic partial differential equations with non-local operator on $\mathbb{R}^{d}$
Yong Zhen Yang, Yong Zhou
TL;DR
The paper develops a unified framework for time-fractional SPDEs on $\mathbb{R}^d$ with Caputo time derivatives and a broad non-local spatial operator $\phi(\Delta)$, driven by Gaussian and Lévy noises. For $1\le p\le 2$ it proves local existence and uniqueness of mild solutions in $L_p$, while for $p>2$ it introduces Besov and Triebel–Lizorkin scales adapted to $\phi(\Delta)$ to obtain solvability and Sobolev regularity of weak solutions. The analysis relies on precise estimates of fundamental solutions, stochastic convolutions, Littlewood–Paley theory, and fixed-point arguments, extending and unifying previous results for fractional SPDEs with non-local operators. The results are valid under a dimensional constraint that ties together $\alpha$, $\sigma_2$, and the growth of the nonlinearities, and they provide a robust toolkit for studying nonlocal temporal-spatial stochastic dynamics with memory and jump effects. Overall, the work broadens the applicability of fractional SPDE techniques to a wide class of non-local operators and Lévy noises, with potential applications in anomalous diffusion and complex media.
Abstract
This paper investigates a class of time-fractional stochastic partial differential equations (SPDEs) on $\mathbb{R}^d$, driven by both Gaussian and Lévy space-time white noises. The equation involves Caputo time-derivatives of orders $α, σ_1, σ_2$ and a non-local operator $φ(Δ)$ generated by a Bernstein function $φ$. We first establish the local existence and uniqueness of mild solutions in the $L_p$-framework for $1 \leq p \leq 2$. For the case $p > 2$, where the $L_p$-theory fails, we develop a Sobolev space framework based on the scales of Besov and Triebel--Lizorkin spaces associated with $φ(Δ)$. Within this setting, we prove the solvability and regularity of weak solutions. Our analysis relies on detailed estimates of the fundamental solutions, stochastic convolutions, Littlewood--Paley theory, and a fixed point argument. The results presented here extend and unify several previous works on fractional SPDEs by incorporating a general class of non-local operators and Lévy noises.