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

We consider the stochastic partial differential equation (SPDE) (Formula presented), where u = u(t, x) is defined for (t, x) ∈ (0, ∞ ) × R and Ẇ 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 measurabl...

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Autores: Foondun, Mohammud, Khoshnevisan, Davar, Nualart, Eulàlia
Tipo de recurso: artículo
Estado:Versión publicada
Fecha de publicación:2026
País:España
Institución:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
Repositorio:Recercat. Dipósit de la Recerca de Catalunya
OAI Identifier:oai:dnet:recercat____::4e04f92f680b95db58a6896f5101cc4e
Acceso en línea:https://hdl.handle.net/10230/73384
http://dx.doi.org/10.1007/s10959-025-01477-y
Access Level:acceso abierto
Palabra clave:SPDEs
Space-time white noise
Existence and uniqueness
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spelling On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficientsFoondun, MohammudKhoshnevisan, DavarNualart, EulàliaSPDEsSpace-time white noiseExistence and uniquenessWe consider the stochastic partial differential equation (SPDE) (Formula presented), where u = u(t, x) is defined for (t, x) ∈ (0, ∞ ) × R and Ẇ 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.Research supported by the Leverhulme Trust Fellowship IF-2025-040, the US-NSF grants DMS-1855439 and DMS-2245242, the Spanish MINECO grant PID2022-138268NB-I00, and Ayudas Fundación BBVA a Proyectos de Investigación Científica 2021.Springer2026202620262026info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionapplication/pdfapplication/pdfhttps://hdl.handle.net/10230/73384http://dx.doi.org/10.1007/s10959-025-01477-yhttps://hdl.handle.net/10230/73384reponame:Recercat. Dipósit de la Recerca de Catalunyainstname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)InglésJournal of Theoretical Probability. 2026 Jun;39(2):23info:eu-repo/grantAgreement/ES/3PE/PID2022-138268NB-I00© The Author(s) 2026. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.http://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:dnet:recercat____::4e04f92f680b95db58a6896f5101cc4e2026-05-29T05:05:01Z
dc.title.none.fl_str_mv On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients
title On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients
spellingShingle On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients
Foondun, Mohammud
SPDEs
Space-time white noise
Existence and uniqueness
title_short On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients
title_full On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients
title_fullStr On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients
title_full_unstemmed On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients
title_sort On the well-posedness of stochastic partial differential equations with locally Lipschitz coefficients
dc.creator.none.fl_str_mv Foondun, Mohammud
Khoshnevisan, Davar
Nualart, Eulàlia
author Foondun, Mohammud
author_facet Foondun, Mohammud
Khoshnevisan, Davar
Nualart, Eulàlia
author_role author
author2 Khoshnevisan, Davar
Nualart, Eulàlia
author2_role author
author
dc.subject.none.fl_str_mv SPDEs
Space-time white noise
Existence and uniqueness
topic SPDEs
Space-time white noise
Existence and uniqueness
description We consider the stochastic partial differential equation (SPDE) (Formula presented), where u = u(t, x) is defined for (t, x) ∈ (0, ∞ ) × R and Ẇ 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.
publishDate 2026
dc.date.none.fl_str_mv 2026
2026
2026
2026
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv https://hdl.handle.net/10230/73384
http://dx.doi.org/10.1007/s10959-025-01477-y
https://hdl.handle.net/10230/73384
url https://hdl.handle.net/10230/73384
http://dx.doi.org/10.1007/s10959-025-01477-y
dc.language.none.fl_str_mv Inglés
language_invalid_str_mv Inglés
dc.relation.none.fl_str_mv Journal of Theoretical Probability. 2026 Jun;39(2):23
info:eu-repo/grantAgreement/ES/3PE/PID2022-138268NB-I00
dc.rights.none.fl_str_mv http://creativecommons.org/licenses/by/4.0/
info:eu-repo/semantics/openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Springer
publisher.none.fl_str_mv Springer
dc.source.none.fl_str_mv reponame:Recercat. Dipósit de la Recerca de Catalunya
instname:Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
instname_str Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya)
reponame_str Recercat. Dipósit de la Recerca de Catalunya
collection Recercat. Dipósit de la Recerca de Catalunya
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