Space-time fractional stochastic partial differential equations driven by Lévy white noise

2025/06/15 by Yuhui Guo, Guo, Yuhui, Jiang-Lun Wu +1 · 1 citation
Economics, Econometrics and Finance · Mathematics · #26A33 #FOS: Mathematics #Fractional Differential Equations Solutions #Primary 60H15 #Probability (math.PR) #Secondary 60G51 #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2506.12834

openalex publication_date 2025/06/15 · openalex created_date 2025/10/14 · openalex updated_date 2026/07/28

Abstract

This paper is concerned with the following space-time fractional stochastic nonlinear partial differential equation (∂_t^β+\fracν2(-Δ)^{α/ 2}) u=I_t^γ[ f(t,x,u)-∑_i=1^d (∂)/(∂ x_i) q_i(t,x,u)+ σ(t,x,u) F_t,x] for a random field u(t,x):[0,∞)×ℝ^d ↦ℝ, where α>0, β∈(0,2), γ≥0, ν>0, F_t,x is a Lévy space-time white noise, I_t^γ stands for the Riemann-Liouville integral in time, and f,q_i,σ:[0,∞)×ℝ^d×ℝ ↦ℝ are measurable functions. Under suitable polynomial growth conditions, we establish the existence and uniqueness of L²(ℝ^d)-valued local solutions when the Lévy white noise F_t,x contains Gaussian noise component. Furthermore, for p∈[1,2], we derive the existence and uniqueness of L^p(ℝ^d)-valued local solutions for the equation driven by pure jump Lévy white noise. Finally, we obtain certain stronger conditions for the existence and uniqueness of global solutions.

Citations

• Sample path properties and small ball probabilities for stochastic fractional diffusion equations
• Stochastic fractional diffusion equations with Gaussian noise rough in space
• Moments and asymptotics for a class of SPDEs with space-time white noise
• Stochastic Burgers' Equation on the Real Line: Regularity and Moment Estimates
• Nonlinear stochastic time-fractional slow and fast diffusion equations on ℝ^d
• Space-time fractional diffusions in Gaussian noisy environment
• Hyperbolicity of minimizers and regularity of viscosity solutions for random Hamilton-Jacobi equations

Cited by

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

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