## Probability Surveys

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

### 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, 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