Equivalences and counterexamples between several definitions of the uniform large deviations principle

TL;DR

This paper investigates various definitions of the uniform large deviations principle, establishes their equivalences and differences through counterexamples, and introduces a new equivalent definition with applications to stochastic equations in Hilbert spaces.

Contribution

It introduces the equicontinuous uniform Laplace principle (EULP) and proves its equivalence to existing definitions, extending the applicability of uniform large deviations principles to infinite-dimensional stochastic equations.

Findings

EULP is equivalent to Freidlin and Wentzell's ULP.

Conditions are provided for functions of Wiener processes to satisfy EULP.

Uniform LDP holds for Hilbert space valued stochastic equations with multiplicative noise.

Abstract

This paper explores the equivalences between four definitions of uniform large deviations principles and uniform Laplace principles found in the literature. Counterexamples are presented to illustrate the differences between these definitions and specific conditions are described under which these definitions are equivalent to each other. A fifth definition called the equicontinuous uniform Laplace principle (EULP) is proposed and proven to be equivalent to Freidlin and Wentzell's definition of a uniform large deviations principle. Sufficient conditions that imply a measurable function of infinite dimensional Wiener process satisfies an EULP using the variational methods of Budhiraja, Dupuis and Maroulas are presented. This theory is applied to prove that a family of Hilbert space valued stochastic equations exposed to multiplicative noise satisfy a uniform large deviations principle that is uniform over all initial conditions in bounded subsets of the Hilbert space. This is an improvement over previous weak convergence methods which can only prove uniformity over compact sets.