On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients

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Foondun, Mohammud and Khoshnevisan, Davar and Nualart, Eulalia (2026) On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients. Journal of Theoretical Probability, 39 (2). 23. ISSN 0894-9840 (https://doi.org/10.1007/s10959-025-01477-y)

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Abstract

We consider the stochastic partial differential equation (SPDE), ∂tu = ½∂ 2x 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 having at most linear growth with regularly behaved local Lipschitz constants. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The novelty of our method is in the pointwise nature of the truncation argument.

ORCID iDs

Foondun, Mohammud ORCID logoORCID: https://orcid.org/0000-0001-7634-916X, Khoshnevisan, Davar and Nualart, Eulalia;
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