Uniform LDP for the SNLSE with combined Stratonovich–Itô noises studied here

Establish a uniform large deviation principle (uniform with respect to initial data, e.g., on bounded subsets of L^2(R^d)) for the family of small‑noise solutions to the stochastic nonlinear Schrödinger equation on R^d with polynomial nonlinearity |u|^{α−1}u, linear damping βu, linear Stratonovich multiplicative noise B(u)∘dW1, and nonlinear Itô multiplicative noise G(u)dW2, as considered in the paper’s model. Specify the uniformity class, topology of the solution space, and the associated good rate function.

Background

The main result establishes an LDP (via Laplace principle) for the considered SNLSE with mixed linear Stratonovich and nonlinear Itô noises under damping, in the solution space C([0,T];L^2) ∩ L^p(0,T;L^r). The proof circumvents missing compact embeddings using a mild formulation and uniform estimates for controlled equations.

They explicitly ask whether one can strengthen this to a uniform LDP under the same general noise structure—an extension that would require uniformity (e.g., over initial data) while retaining the challenging mixed noise and dispersive–nonlinear setting.