Probability Surveys

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

Michael Salins

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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.

Article information

Source
Probab. Surveys, Volume 16 (2019), 99-142.

Dates
Received: June 2018
First available in Project Euclid: 31 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.ps/1559289659

Digital Object Identifier
doi:10.1214/18-PS309

Mathematical Reviews number (MathSciNet)
MR3960292

Zentralblatt MATH identifier
07064383

Subjects
Primary: 60F10: Large deviations 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
Large deviations uniform large deviations stochastic processes stochastic partial differential equations

Rights
Creative Commons Attribution 4.0 International License.

Citation

Salins, Michael. Equivalences and counterexamples between several definitions of the uniform large deviations principle. Probab. Surveys 16 (2019), 99--142. doi:10.1214/18-PS309. https://projecteuclid.org/euclid.ps/1559289659


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