Large deviation principle for a stochastic nonlinear damped Schrodinger equation

TL;DR

The article establishes a large deviation principle for solutions to a stochastic nonlinear Schrödinger equation with polynomial nonlinearity and mixed Stratonovich/Itô noise, in the presence of linear damping. By recasting the Stratonovich term into Itô form and employing a weak convergence (Budhiraja–Dupuis) framework, the authors prove the Laplace principle in the space $\mathcal{E}=C([0,T];H)\cap L^p(0,T;L^r(\mathbb{R}^d))$, with a rate function defined via a skeleton (deterministic controlled) equation. The main technical contributions include proving global well-posedness of the skeleton equation using Banach fixed point and Yosida approximations, constructing and controlling a truncated stochastic controlled equation, and establishing uniform bounds to verify the weak convergence criterion. The work overcomes challenges posed by the lack of compact embeddings and the absence of conservation laws (except in a special case), enabling LDP results in full solution spaces for arbitrary dimensions and subcritical nonlinearities. Overall, the paper provides a rigorous probabilistic description of rare events for nonlinear dispersive dynamics under mixed stochastic perturbations and paves the way for future extensions to broader noise structures and rough paths. The results have potential implications for understanding stability and rare transition phenomena in random dispersive systems.

Abstract

The present paper focuses on the stochastic nonlinear Schrodinger equation with polynomial nonlinearity, and a zero-order (no derivatives involved) linear damping. Here, the random forcing term appears as a mix of a nonlinear noise in the Ito sense and a linear multiplicative noise in the Stratonovich sense. We prove the Laplace principle for the family of solutions to the stochastic system in a suitable Polish space, using the weak convergence framework of Budhiraja and Dupuis. This analysis is nontrivial, since it requires uniform estimates for the solutions of the associated controlled stochastic equation in the underlying solution space in order to verify the weak convergence criterion. The Wentzell Freidlin type large deviation principle is proved using Varadhan's lemma and Bryc's converse to Varadhan's lemma. The local well-posedness of the skeleton equation (deterministic controlled system) is established by employing the Banach fixed point theorem, and the global well posedness is established via Yosida approximation. We show that the conservation law holds in the absence of the linear damping and Ito noise. The well posedness of the stochastic controlled equation is also nontrivial in this case. We use a truncation method, a stopping time argument, and the Yosida technique to get the global well-posedness of the stochastic controlled equation.

Theorems & Definitions (49)

• Definition 2.1: Rate function
• Definition 2.2: Large deviation principle
• Definition 2.3: Laplace principle
• Lemma 2.1: Varadhan’s Lemma MR758258
• Lemma 2.2: Bryc’s Lemma MR1431744
• Lemma 2.3
• Theorem 2.1
• Definition 3.1: Mild solution
• Definition 3.2: Admissible pair
• Proposition 3.1: Strichartz's estimates