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Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise

Weina Wu, Jianliang Zhai, Jiahui Zhu

Abstract

We establish a Freidlin-Wentzell type large deviation principle (LDP) for a class of stochastic partial differential equations with locally monotone coefficients driven by Lévy noise. Our results essentially improve a recent work on this topic (Bernoulli, 2018) by the second named author of this paper and his collaborator, because we drop the compactness embedding assumptions, and we also make the conditions for the coefficient of the noise term more specific and weaker. To obtain our results, we utilize an improved sufficient criteria of Budhiraja, Chen, Dupuis, and Maroulas for functions of Poisson random measures, and the techniques introduced by the first and second named authors of this paper in \cite{WZSIAM} play important roles. As an application, for the first time, the Freidlin-Wentzell type LDPs for many SPDEs driven by Lévy noise in unbounded domains of $\mathbb{R}^d$, which are generally lack of compactness embeddings properties, are achieved, like e.g., stochastic $p$-Laplace equation, stochastic Burgers-type equations, stochastic 2D Navier-Stokes equations, stochastic equations of non-Newtonian fluids, etc.

Large deviations for locally monotone stochastic partial differential equations driven by Lévy noise

Abstract

We establish a Freidlin-Wentzell type large deviation principle (LDP) for a class of stochastic partial differential equations with locally monotone coefficients driven by Lévy noise. Our results essentially improve a recent work on this topic (Bernoulli, 2018) by the second named author of this paper and his collaborator, because we drop the compactness embedding assumptions, and we also make the conditions for the coefficient of the noise term more specific and weaker. To obtain our results, we utilize an improved sufficient criteria of Budhiraja, Chen, Dupuis, and Maroulas for functions of Poisson random measures, and the techniques introduced by the first and second named authors of this paper in \cite{WZSIAM} play important roles. As an application, for the first time, the Freidlin-Wentzell type LDPs for many SPDEs driven by Lévy noise in unbounded domains of , which are generally lack of compactness embeddings properties, are achieved, like e.g., stochastic -Laplace equation, stochastic Burgers-type equations, stochastic 2D Navier-Stokes equations, stochastic equations of non-Newtonian fluids, etc.
Paper Structure (6 sections, 6 theorems, 124 equations)

This paper contains 6 sections, 6 theorems, 124 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hypotheses and main result
  4. Well-posedness for the skeleton equation
  5. The verification of (LDP1)
  6. The verification of (LDP2)

Key Result

Proposition 3.2

Suppose that conditions $\textbf{(H1)}$-$\textbf{(H7)}$ hold. Then eq SPDE 01 has a unique solution $X^\epsilon=\{X^\epsilon_t\}_{t\in[0,T]}$.

Theorems & Definitions (19)

  • Definition 2.1
  • Remark 2.2
  • Remark 3.1
  • proof
  • Proposition 3.2
  • Remark 3.3
  • Remark 3.4
  • Definition 3.5
  • Definition 3.6
  • Theorem 3.7
  • ...and 9 more