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

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.

💡 Research Summary

The paper establishes a uniform Donsker‑Varadhan large deviation principle (LDP) for the empirical measures of solutions to a randomly forced nonlinear Schrödinger (NLS) equation on the one‑dimensional torus. The equation includes a smooth, non‑negative damping term a(x), a power‑type nonlinearity |u|^{p‑1}u with odd exponent p≥3, and a stochastic forcing η(t,x) that is colored in space, periodic in time, and acts only on finitely many Fourier modes (the “determining modes”). The noise is constructed as a sequence of i.i.d. kicks on each unit time interval, with coefficients b_{j,k} satisfying a weighted ℓ² bound (1.2) and vanishing for low frequencies (1.3). Under these structural assumptions the deterministic flow map S(u₀,ζ)=u(1) is locally Lipschitz on H¹(T), and the discrete‑time random dynamical system (RDS) u_{n+1}=S(u_n,η_n) fits into a general framework of Markov chains on a compact metric space X.

The authors introduce a new abstract LDP criterion (Theorem 2.2) for such RDS. The criterion requires two properties: (I) uniform irreducibility – any two points can be connected with uniformly positive probability in a uniformly bounded number of steps; and (C) a coupling (squeezing) condition – there exists a fixed contraction factor q∈