Nonlinear stochastic time-fractional slow and fast diffusion equations on ℝ^d

2015/09/25 by Le Chen, Chen, Le, Yaozhong Hu +3 · 1 citation
Economics, Econometrics and Finance · Mathematics · #35R60 #60G60 #60H15 #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:35R60 #msc:60G60 #msc:60H15

paper · pdf · doi:10.48550/arxiv.1509.07763

43 pages, 4 figures

arxiv created 2015/09/25 · openalex publication_date 2015/09/25 · arxiv updated 2015/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: (∂^β+\fracν2(-Δ)^α/2)u(t,x) = I_t^γ[ρ(u(t,x))W(t,x)], t>0, x∈ℝ^d, where W is the space-time white noise, α∈(0,2], β∈(0,2), γ≥ 0 and ν>0. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang's condition: d<2α+\fracαβmin(2γ-1,0). In some cases, the initial data can be measures. When β∈ (0,1], we prove the sample path regularity of the solution.

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

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