The time fractional stochastic partial differential equations with non-local operator on ℝ^d

2025/12/03 by Yong Zhen Yang, Yong Zhou, Yang, Yong Zhen +1
Economics, Econometrics and Finance · Mathematics · #26A33 #35R11 #Analysis of PDEs (math.AP) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2512.03754

openalex publication_date 2025/12/03 · openalex created_date 2025/12/05 · openalex updated_date 2026/07/28

Abstract

This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on ℝ^d driven by mixed Wiener--Lévy noises. The equations feature a Caputo time derivative ∂_t^α (0<α<1) and a spatial nonlocal operator ϕ(Δ) generated by a subordinate Brownian motion, leading to a doubly nonlocal structure. For the case p ≥ 2, we prove the existence, uniqueness, and sharp Sobolev regularity of weak solutions in the scale of ϕ-Sobolev spaces H_p^ϕ,γ+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 < 2κ₀(2 - (2σ₂ - 2/p)₊/α) is obtained.~For the low-regularity case 1 ≤ p ≤ 2, where maximal function estimates fail, we construct unique local mild solutions in L_p(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.

Citations

• Space-time fractional stochastic partial differential equations driven by Lévy white noise
• A Sobolev space theory for the Stochastic Partial Differential Equations with space-time non-local operators
• An L_q(L_p)-theory for diffusion equations with space-time nonlocal operators
• On the Cauchy problem for integro-differential equations in the scale of spaces of generalized smoothness
• A Sobolev Space theory for stochastic partial differential equations with time-fractional derivatives
• Fractional time stochastic partial differential equations
• Parabolic Littlewood-Paley inequality for ϕ(-Δ)-type operators and applications to Stochastic integro-differential equations
• An L^p-theory of non-divergence form SPDEs driven by Lévy processes
• Cylindrical Lévy processes in Banach spaces
• Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications

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• Fractional SPDEs driven by spatially correlated noise: existence of the solution and smoothness of its density