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Dynamics and large deviations for fractional stochastic partial differential equations with Lévy noise

Jiaohui Xu, Tomás Caraballo, José Valero

TL;DR

The paper analyzes fractional SPDEs on bounded domains driven by Lévy noise and Brownian motion, formulating a nonlocal model with the fractional Laplacian and nonlinearities. It develops a comprehensive framework proving well-posedness, weak mean random attractors, invariant measures with ergodicity, and a large deviation principle via a variational representation for Poisson and Brownian inputs, with a concrete Chafee–Infante example illustrating the results. The contributions advance the theory of nonlocal stochastic dynamics with jumps, providing rigorous long-time behavior results and a principled LDP methodology for infinite-dimensional systems. The work combines energy methods, compactness arguments, and variational large-deviation techniques to link stochastic dynamics, attractors, invariant measures, and rare-event analysis in fractional SPDEs.

Abstract

This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by Lévy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then, the existence and uniqueness of weak pullback mean random attractors for the equations {are} established by defining a mean random dynamical system. Next, we prove the existence of invariant measures when the problem is autonomous by means of the fact that $H^γ(\mathcal{O})$ is compactly embedded in $L^2(\mathcal{O})$ with $γ\in (0,1)$. Moreover, the uniqueness of this invariant measure is presented which ensures the ergodicity of the problem. Finally, a large deviation principle result for solutions of SPDEs perturbed by small Lévy noise and Brownian motion is obtained by a variational formula for positive functionals of a Poisson random measure and Brownian motion. Additionally, the results are illustrated by the fractional stochastic Chafee-Infante equations

Dynamics and large deviations for fractional stochastic partial differential equations with Lévy noise

TL;DR

The paper analyzes fractional SPDEs on bounded domains driven by Lévy noise and Brownian motion, formulating a nonlocal model with the fractional Laplacian and nonlinearities. It develops a comprehensive framework proving well-posedness, weak mean random attractors, invariant measures with ergodicity, and a large deviation principle via a variational representation for Poisson and Brownian inputs, with a concrete Chafee–Infante example illustrating the results. The contributions advance the theory of nonlocal stochastic dynamics with jumps, providing rigorous long-time behavior results and a principled LDP methodology for infinite-dimensional systems. The work combines energy methods, compactness arguments, and variational large-deviation techniques to link stochastic dynamics, attractors, invariant measures, and rare-event analysis in fractional SPDEs.

Abstract

This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by Lévy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then, the existence and uniqueness of weak pullback mean random attractors for the equations {are} established by defining a mean random dynamical system. Next, we prove the existence of invariant measures when the problem is autonomous by means of the fact that is compactly embedded in with . Moreover, the uniqueness of this invariant measure is presented which ensures the ergodicity of the problem. Finally, a large deviation principle result for solutions of SPDEs perturbed by small Lévy noise and Brownian motion is obtained by a variational formula for positive functionals of a Poisson random measure and Brownian motion. Additionally, the results are illustrated by the fractional stochastic Chafee-Infante equations
Paper Structure (19 sections, 25 theorems, 228 equations)

This paper contains 19 sections, 25 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractional setting
  4. Stochastic setting and assumptions
  5. Large deviation principle
  6. Well-posedness of problem \ref{['eq2-13']}-\ref{['eq2-14']}
  7. Existence of weak mean random attractors
  8. Invariant measures and ergodicity
  9. Introduction to invariant measures and ergodicity
  10. Existence of invariant measures
  11. Ergodicity
  12. Large deviation Principle
  13. Controlled Poisson random measure
  14. Poisson random measure and Brownian motion
  15. A general criterium
  16. ...and 4 more sections

Key Result

Theorem 3.2

Assume conditions $(F_1)$-$(F_3)$, $(g_1)$-$(g_3)$ and $(h_1)$-$(h_3)$ hold. Then, for every $\mathcal{F}_\tau$-measurable initial value $u_0\in L^2(\Omega;\mathbb{H})$, problem eq2-13-eq2-14 has a unique global solution in the sense of Definition def3-1. Moreover, the solution $u$ depends continuou

Theorems & Definitions (32)

  • Definition 3.1
  • Theorem 3.2
  • Remark 3.3
  • Lemma 4.1
  • Lemma 4.2
  • Theorem 4.3
  • Definition 5.1
  • Definition 5.2
  • Lemma 5.3
  • Lemma 5.4
  • ...and 22 more