Space-time fractional SPDEs with locally Lipschitz coefficients: well-posedness

2025/11/09 by Ngartelbaye Guerngar, Guerngar, Ngartelbaye, Erkan Nane +1
Economics, Econometrics and Finance · Mathematics · #60H15 #60H30 #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Differential Equations Analysis #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2511.06541

openalex publication_date 2025/11/09 · openalex created_date 2025/11/12 · openalex updated_date 2026/07/28

Abstract

In this article, we study the space-time SPDE ∂_t^βu=-(-Δ)^α/2 u+I_t^1-β[b(u)+σ(u)W], where u=u(t,x) is defined for (t,x)∈ℝ₊× ℝ, β∈(0,1), α∈(0,2) and W denotes a space-time white noise. It has long been conjectured that this equation has a unique solution with finite moments under the minimal assumptions of locally Lipschitz coefficients b and σ with linear growth. We prove that this SPDE is well-posed under the assumptions that the initial condition u₀ is bounded and measurable, and the functions b and σ are locally Lipschitz and have at-most linear growth and some conditions on the Lipschitz constants on the truncated versions of b and σ. Our results generalize the work of Foondun et al.(2025) to a space-time fractional setting.

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