On the well-posedness of SPDEs with locally Lipschitz coefficients

Mohammud Foondun; Davar Khoshnevisan; Eulalia Nualart

doi:10.48550/arXiv.2411.09381

On the well-posedness of SPDEs with locally Lipschitz coefficients

Mohammud Foondun
, Davar Khoshnevisan
, Eulalia Nualart

Mathematics And Statistics

Research output: Working paper/Preprint/Pre-registration › Working Paper/Preprint

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Abstract

We consider the stochastic partial differential equation, ∂_tu = ½∂²_xu+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.

Original language	English
Place of Publication	Ithaca, NY
Pages	17
DOIs	https://doi.org/10.48550/arXiv.2411.09381
Publication status	Published - 14 Nov 2024

Funding

Research supported by the Leverhulme Trust Fellowship IF-2025-040, the US-NSF grants DMS1855439 and DMS-2245242, the Spanish MINECO grant PID2022-138268NB-100, and Ayudas Fundacion BBVA a Proyectos de Investigación Científica 2021.

Keywords

SPDEs
space-time white noise
existence and uniqueness

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10.48550/arXiv.2411.09381Licence: Other

Foondun-etal-arXiv-2024-On-the-well-posedness-of-SPDEs-with-locally-Lipschitz-coefficientsFinal published version, 592 KBLicence: Strath_1

1 Article

On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients
Foondun, M., Khoshnevisan, D. & Nualart, E., 23 Feb 2026, In: Journal of Theoretical Probability. 39, 2, 21 p., 23.
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abstract = "We consider the stochastic partial differential equation, ∂tu = ½∂2xu+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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N2 - We consider the stochastic partial differential equation, ∂tu = ½∂2xu+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.

AB - We consider the stochastic partial differential equation, ∂tu = ½∂2xu+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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BT - On the well-posedness of SPDEs with locally Lipschitz coefficients

CY - Ithaca, NY

ER -

On the well-posedness of SPDEs with locally Lipschitz coefficients

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

Cite this