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, and under stronger assumptions is uniform over initial conditions in unbounded subsets too. This is an improvement over previous weak convergence methods which can only prove uniformity over compact sets.
Highlights
The theory of large deviations principles, developed in the 1960s by Freidlin, Wentzell, Varadhan and others, characterizes the asymptotic decay rate of rare probabilities
We propose the new definition of the equicontinuous uniform Laplace principle (EULP) (Definition 2.9 below)
We show that if B and G are globally Lipschitz continuous in an appropriate sense, the mild solutions to Xxε satisfy a EULP in E = C([0, T ] : H) that is uniform over initial conditions in bounded subsets of H
Summary
The theory of large deviations principles, developed in the 1960s by Freidlin, Wentzell, Varadhan and others, characterizes the asymptotic decay rate of rare probabilities. When studying the exit time of Xxε from a bounded domain, it is sufficient to prove uniformity of the large deviations principle over initial conditions in compact sets because all closed bounded sets are compact. The pre-compactness of x∈A Φx(s) is required to prove the equivalence between the FWULDP and the ULP precisely because continuous functions on compact the case fsoertsXaxεr(et)u=nifxor+m√lyεcWon(tti)nuwoituhs.AW=heRn, this compactness is lacking, as is we build our counterexample by choosing a function h : E → R that is continuous, but not uniformly continuous Based on this observation, we propose the new definition of the equicontinuous uniform Laplace principle (EULP) (Definition 2.9 below). Appendices B and C include some proofs about the Hilbert space valued process from Section 4
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