Well-posedness of stochastic evolution equations with Hölder continuous noise

We show existence and pathwise uniqueness of probabilistically strong solutions to a pseudomonotone stochastic evolution problem on a bounded domain \(D\subseteq\mathbb{R}^d\), \(d\in\mathbb{N}\), with homogeneous Dirichlet boundary conditions and random initial data \(u_0\in L^2(\Omega;L^2(D))\). T...

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Bibliographic Details
Published inarXiv.org
Main Authors Schmitz, Kerstin, Zimmermann, Aleksandra
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 18.03.2024
Subjects
Boundary conditions
Continuous noise
Dirichlet problem
Evolution
Martingales
Uniqueness
Well posed problems
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Summary:We show existence and pathwise uniqueness of probabilistically strong solutions to a pseudomonotone stochastic evolution problem on a bounded domain \(D\subseteq\mathbb{R}^d\), \(d\in\mathbb{N}\), with homogeneous Dirichlet boundary conditions and random initial data \(u_0\in L^2(\Omega;L^2(D))\). The main novelty is the presence of a merely H\"older continuous multiplicative noise term. In order to show the well-posedness, we simultaneously regularize the H\"older noise term by inf-convolution and add a perturbation by a higher order operator to the equation. Using a stochastic compactness argument we may pass to the limit and we obtain first a martingale solution. Then by a pathwise uniqueness argument we get existence of a probabilistically strong solution.
ISSN:2331-8422