LDP for generalized (non‑Gaussian/fractional) stochastic Schrödinger systems

Determine whether a large deviation principle holds for generalized stochastic nonlinear Schrödinger equations obtained by replacing Gaussian driving noise with non‑Gaussian laws such as Lévy α‑stable processes, including time‑ or space‑fractional Schrödinger formulations. Concretely, establish an LDP for the small‑noise family of solutions to such non‑Gaussian, possibly fractional, Schrödinger dynamics on R^d with polynomial nonlinearity |u|^{α−1}u, identifying an appropriate state space and good rate function.

Background

The paper proves a Freidlin–Wentzell type large deviation principle for a stochastic nonlinear Schrödinger equation on R^d with polynomial nonlinearity, linear damping, linear Stratonovich multiplicative noise via a bounded operator B, and nonlinear Itô multiplicative noise via a Lipschitz G. The analysis uses Strichartz/smoothing estimates and Yosida approximations to handle the lack of compactness.

The authors note that, in a path‑integral perspective, replacing Gaussian measures with non‑Gaussian (e.g., Lévy α‑stable) distributions leads to generalized Schrödinger equations, including time‑ or space‑fractional variants that capture heavy‑tailed jump statistics. They explicitly raise the question of whether an LDP can be developed for these generalized (non‑Gaussian/fractional) systems.