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Well-posedness of stochastic evolution equations with Hölder continuous noise

Kerstin Schmitz, Aleksandra Zimmermann

Abstract

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 $u_0\in L^2(Ω;L^2(D))$. The main novelty is the presence of a merely Hölder continuous multiplicative noise term. In order to show the well-posedness, we simultaneously regularize the Hölder 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.

Well-posedness of stochastic evolution equations with Hölder continuous noise

Abstract

We show existence and pathwise uniqueness of probabilistically strong solutions to a pseudomonotone stochastic evolution problem on a bounded domain , , with homogeneous Dirichlet boundary conditions and random initial data . The main novelty is the presence of a merely Hölder continuous multiplicative noise term. In order to show the well-posedness, we simultaneously regularize the Hölder 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.
Paper Structure (13 sections, 26 theorems, 141 equations)

This paper contains 13 sections, 26 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Statement of the problem, motivation, and former results
  3. Hypotheses
  4. Main results and outline
  5. Existence of approximate solutions
  6. Lipschitz continuous approximation of the noise
  7. A higher order perturbation
  8. Well-posedness of the approximated equation
  9. Existence of a martingale solution
  10. A priori estimates
  11. Tightness results
  12. Passage to the limit
  13. Pathwise uniqueness of solutions

Key Result

Theorem 1.4

Assume that $f\in L^\infty(\mathbb{R})$ is a given Lipschitz continuous function, and $\sigma:(0,T)\times\mathbb{R}\rightarrow\mathbb{R}$ fulfills (S1)-(S3) for an arbitrary $\alpha\in(0,1)$. Then, spde admits a martingale solution in the sense of Definition defsolution$ii)$.

Theorems & Definitions (35)

  • Remark 1.1
  • Remark 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Definition 2.1
  • Proposition 2.2
  • Corollary 2.3
  • ...and 25 more