Uniform large deviation principles for Banach space valued stochastic differential equations

TL;DR

This paper establishes a uniform large deviation principle for Banach space valued stochastic differential equations, enabling analysis of exit times and asymptotic behaviors in infinite-dimensional stochastic systems.

Contribution

It introduces a novel approach to prove uniform LDPs for Banach space SDEs by embedding into the double dual and utilizing weak-* compactness, extending finite-dimensional results.

Findings

Proves a uniform LDP for Banach space valued SDEs.

Applies the uniform LDP to reaction diffusion and Navier-Stokes equations.

Provides insights into exit time asymptotics for infinite-dimensional stochastic systems.

Abstract

We prove a large deviation principle (LDP) for a general class of Banach space valued stochastic differential equations (SDE) that is uniform with respect to initial conditions in bounded subsets of the Banach space. A key step in the proof is showing that a uniform large deviation principle over compact sets is implied by a uniform over compact sets Laplace principle. Because bounded subsets of infinite dimensional Banach spaces are in general not relatively compact in the norm topology, we embed the Banach space into its double dual and utilize the weak-$\star $ compactness of closed bounded sets in the double dual space. We prove that a modified version of our stochastic differential equation satisfies a uniform Laplace principle over weak-$\star $ compact sets and consequently a uniform over bounded sets large deviation principle. We then transfer this result back to the original equation using a contraction principle. The main motivation for this uniform LDP is to generalize results of Freidlin and Wentzell concerning the behavior of finite dimensional SDEs. Here we apply the uniform LDP to study the asymptotics of exit times from bounded sets of Banach space valued small noise SDE, including reaction diffusion equations with multiplicative noise and $2$-dimensional stochastic Navier-Stokes equations with multiplicative noise.