Small time asymptotics for a class of stochastic partial differential equations with fully monotone coefficients forced by multiplicative Gaussian noise

TL;DR

This paper establishes small time large deviation principles for a broad class of stochastic partial differential equations with monotone coefficients driven by multiplicative Gaussian noise, covering many important models.

Contribution

It provides a unified framework for small time asymptotics of various SPDEs with nonlinear drifts and multiplicative noise, extending previous results to more general equations.

Findings

Derived small time large deviation principles for SPDEs.

Applicable to models like Navier-Stokes, Cahn-Hilliard, and porous medium equations.

Demonstrated the effect of nonlinear drifts under multiplicative Gaussian noise.

Abstract

The main goal of this article is to study the effect of small, highly nonlinear, unbounded drifts (small time large deviation principle (LDP) based on exponential equivalence arguments) for a class of stochastic partial differential equations (SPDEs) with fully monotone coefficients driven by multiplicative Gaussian noise. The small time LDP obtained in this paper is applicable for various quasi-linear and semilinear SPDEs such as porous medium equations, Cahn-Hilliard equation, 2D Navier-Stokes equations, convection-diffusion equation, 2D liquid crystal model, power law fluids, Ladyzhenskaya model, $p$-Laplacian equations, etc., perturbed by multiplicative Gaussian noise.