On the well-posedness of SPDEs with locally Lipschitz coefficients

2024/11/14 by Mohammud Foondun, Davar Khoshnevisan, Foondun, Mohammud +3 · 2 citations
Computer Science · Economics, Econometrics and Finance · Engineering · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Reservoir Engineering and Simulation Methods #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2411.09381

openalex publication_date 2024/11/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the stochastic partial differential equation, ∂_t u = \tfrac12 ∂²_x u + b(u) + σ(u) W, where u=u(t ,x) is defined for (t ,x)∈(0 ,∞)×ℝ, and W denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition u(0) is bounded and measurable, and b and σ are locally Lipschitz continuous functions and have at most linear growth. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The results naturally generalize to the case where b and σ are time dependent with uniform-in-time growth and oscillation properties. Additionally, our method can be extended to the stochastic wave equation.

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