Large deviation principle for a stochastic nonlinear damped Schrodinger equation

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.

💡 Research Summary

The paper studies a stochastic nonlinear Schrödinger equation (SNLSE) with a polynomial nonlinearity, a zero‑order linear damping term, and a mixed noise consisting of a linear multiplicative Stratonovich component and a nonlinear Itô component. The authors aim to establish a Freidlin‑Wentzell type large deviation principle (LDP) for the family of solutions as the noise intensity ε→0, using the weak convergence framework of Budhiraja and Dupuis.

The model is written as
du(t)=−iA u(t) dt + iN(u(t)) dt + β u(t) dt − i√ε B(u(t))∘dW₁(t) − i√ε G(u(t)) dW₂(t),
with A a non‑negative self‑adjoint operator, β≥0 the damping coefficient, B a bounded linear operator, N(u)=|u|^{α‑1}u (α>1), and G a nonlinear operator. W₁ and W₂ are independent Wiener processes; W₁ appears in Stratonovich form, W₂ in Itô form.

The analysis proceeds in several stages. First, the deterministic “skeleton” equation obtained by replacing the stochastic terms with a control h∈L²(0,T;H) is studied. By defining an integral operator on a suitable Banach space and showing it is a strict contraction, the Banach fixed‑point theorem yields a unique local mild solution. Global existence follows from a Yosida approximation of the Schrödinger operator together with Strichartz and local smoothing estimates, which compensate for the lack of intrinsic regularisation of the Schrödinger flow. The authors also discuss the special conservative case (β=0, G≡0) where the L²‑norm (mass) is preserved.

Next, the stochastic controlled equation (the original SNLSE with an added deterministic control term) is tackled. Because the nonlinear noise prevents direct global estimates, the authors introduce a truncation function χ_R and a stopping time τ_R that records when the solution leaves a bounded L^r‑norm region. On the interval