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Donsker-Varadhan large deviation principle for locally damped and randomly forced NLS equations

Yuxuan Chen, Shengquan Xiang

TL;DR

This work establishes a Donsker–Varadhan large deviation principle for randomly forced PDEs with damping and degenerate noise acting on finite modes. It introduces a general LDP criterion for random dynamical systems, built on irreducibility and a fixed-rate squeezing condition, and overcomes the challenge of a non-vanishing squeezing rate with a novel bootstrap argument for Lipschitz bounds on Feynman–Kac semigroups. The authors verify the criterion in three settings: a nonlinear Schrödinger equation on $ ext{1D torus}$, a nonlinear wave equation, and the two-dimensional Navier–Stokes system with degenerate noise, obtaining uniform LDPs for empirical measures on the support of the unique invariant measure $ mu_*$. The results pave the way for LDP analyses of broader dissipative PDEs with degenerate forcing and offer a pathway to studying observables via the contraction principle. They also discuss the limitations related to initial data outside $ ext{supp}( mu_*)$ and outline future work on hyperbolic equations with white-in-time noise.

Abstract

We study large deviations from the invariant measure for nonlinear Schrödinger equations with colored noises on determining modes. The proof is based on a new abstract criterion, inspired by [V. Jakšić et al., Comm. Pure Appl. Math., 68 (2015), 2108-2143]. To address the difficulty caused by fixed squeezing rate, we introduce a bootstrap argument to derive Lipschitz estimates for Feynman-Kac semigroups. This criterion is also applicable to wave equations and Navier-Stokes system.

Donsker-Varadhan large deviation principle for locally damped and randomly forced NLS equations

TL;DR

This work establishes a Donsker–Varadhan large deviation principle for randomly forced PDEs with damping and degenerate noise acting on finite modes. It introduces a general LDP criterion for random dynamical systems, built on irreducibility and a fixed-rate squeezing condition, and overcomes the challenge of a non-vanishing squeezing rate with a novel bootstrap argument for Lipschitz bounds on Feynman–Kac semigroups. The authors verify the criterion in three settings: a nonlinear Schrödinger equation on , a nonlinear wave equation, and the two-dimensional Navier–Stokes system with degenerate noise, obtaining uniform LDPs for empirical measures on the support of the unique invariant measure . The results pave the way for LDP analyses of broader dissipative PDEs with degenerate forcing and offer a pathway to studying observables via the contraction principle. They also discuss the limitations related to initial data outside and outline future work on hyperbolic equations with white-in-time noise.

Abstract

We study large deviations from the invariant measure for nonlinear Schrödinger equations with colored noises on determining modes. The proof is based on a new abstract criterion, inspired by [V. Jakšić et al., Comm. Pure Appl. Math., 68 (2015), 2108-2143]. To address the difficulty caused by fixed squeezing rate, we introduce a bootstrap argument to derive Lipschitz estimates for Feynman-Kac semigroups. This criterion is also applicable to wave equations and Navier-Stokes system.

Paper Structure

This paper contains 16 sections, 14 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Literature review
  4. New ingredient
  5. Organization
  6. Notations.
  7. Model and results
  8. RDS model
  9. Proof of the Main Theorem
  10. Two more applications
  11. Nonlinear wave equation
  12. Navier--Stokes system
  13. Proof for the LDP criterion
  14. Some reductions
  15. Lipschitz estimates for $Q_n^V$
  16. ...and 1 more sections

Key Result

Proposition 2.1

Under the above settings, suppose that hypotheses $(\mathbf{I})$ and $(\mathbf{C})$ are satisfied. Then the Markov process $(x_n,\mathbb{P}_x)$ defined by RDS admits a unique invariant measure $\mu_*\in\mathcal{P}(X)$. Moreover, $\operatorname{supp}(\mu_*)=X$, and there exist constants $C,\gamma>0$

Theorems & Definitions (35)

  • Example 1.1: Parabolic equation with degenerate noise
  • Example 1.2: Hyperbolic equation
  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4: Attainable set
  • proof : Proof of the Main Theorem
  • proof : Proof of the Main Theorem
  • Definition 2.5: $\Gamma$-type domain
  • Theorem 2.6
  • ...and 25 more