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Existence and regularity of random attractors for stochastic evolution equations driven by rough noise

Alexandra Neamtu, Tim Seitz

Abstract

This work establishes the existence and regularity of random pullback attractors for parabolic partial differential equations with rough nonlinear multiplicative noise under natural assumptions on the coefficients. To this aim, we combine tools from rough path theory and random dynamical systems.~An application is given by partial differential equations with rough boundary noise, for which flow transformations are not available.

Existence and regularity of random attractors for stochastic evolution equations driven by rough noise

Abstract

This work establishes the existence and regularity of random pullback attractors for parabolic partial differential equations with rough nonlinear multiplicative noise under natural assumptions on the coefficients. To this aim, we combine tools from rough path theory and random dynamical systems.~An application is given by partial differential equations with rough boundary noise, for which flow transformations are not available.
Paper Structure (9 sections, 19 theorems, 100 equations)

This paper contains 9 sections, 19 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Rough path theory
  4. Random dynamics
  5. Gronwall lemmata and the Mittag-Leffler function
  6. Random attractor
  7. Applications
  8. PDEs with multiplicative rough boundary noise
  9. Parabolic PDEs with multiplicative rough noise

Key Result

Theorem 2.6

(GHN2021). Let $(y,y')\in \mathcal{D}^{2\gamma}_{X,\alpha}([s,t])$, then the limit exists as an element in $E_{\alpha-2\gamma}$, where $\mathcal{P}$ denotes a partition of $[s,t]$. For $0\leq \beta< 3\gamma$ the following estimate holds true, where $C_I:=C_I(\alpha,\gamma,\beta)>0$.

Theorems & Definitions (50)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.5
  • Theorem 2.6
  • Lemma 2.8
  • Definition 2.9
  • Definition 2.10
  • Definition 2.11
  • Lemma 2.12
  • ...and 40 more