Large Deviations for Stochastic equations in Hilbert Spaces with non-Lipschitz drift

TL;DR

This paper establishes a large deviations principle for infinite-dimensional stochastic differential equations with non-Lipschitz continuous drifts, broadening the scope of stochastic analysis in Hilbert spaces.

Contribution

It extends Freidlin-Wentzell theory to Hilbert space SDEs with continuous, non-Lipschitz drifts, including nonlinear fractional diffusion equations with white noise.

Findings

Proves a large deviations principle for non-Lipschitz SDEs in Hilbert spaces.

Applies to a wide class of nonlinear fractional diffusion equations.

Handles cylindrical Wiener process perturbations without Lipschitz assumptions.

Abstract

We prove a Freidlin-Wentzell result for stochastic differential equations in infinite-dimensional Hilbert spaces perturbed by a cylindrical Wiener process. We do not assume the drift to be Lipschitz continuous, but only continuous with at most linear growth. Our result applies, in particular, to a large class of nonlinear fractional diffusion equations perturbed by a space-time white noise.