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

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.

Large deviations for fast transport stochastic RDEs with applications to the exit problem

“…The following proposition of [18] then shows that the uniform Laplace principle implies the uniform large deviation principle. Proposition 2.7.…”

“…The equivalence between the non-uniform versions of the large deviation principle and the Laplace principle is a well known fact. Recently, in [18] general results on the equivalence between the uniform versions of the large deviation principle and the Laplace principle have been investigated.…”

“…These arguments can be extended in our setting to give uniformity with respect to initial conditions in bounded sets. Thus, our job here is proving that the conditions introduced in [18] are satisfied.…”

“…More delicate is the problem of understanding how the uniform Laplace principle may imply the uniform large deviation principle. To this purpose, recently, in [18], some conditions have been introduced in order to guarantee, among other things, the equivalence between the uniform Laplace principle and the uniform large deviation principle, with respect to initial conditions in a compact set. These arguments can be extended in our setting to give uniformity with respect to initial conditions in bounded sets.…”

Large deviations principle for the invariant measures of the 2D stochastic Navier-Stokes equations with vanishing noise correlation

“…In the proof of the main result, Theorem 2.1, two different non-equivalent formulations of the uniform large deviations principle will be required. A thorough comparative analysis of the different formulations of uniform LDPs is given in the paper [16]. We state the definitions for the two forms needed in this paper, using the same notations and conventions as in [16].…”