Large deviations for infinite dimensional stochastic dynamical systems

TL;DR

This paper develops a new approach to large deviations analysis for infinite dimensional stochastic systems driven by Brownian noise, avoiding complex approximations by using a variational representation.

Contribution

It introduces a method that bypasses approximation constructions in large deviations analysis for infinite dimensional stochastic differential equations.

Findings

Simplifies large deviations proofs for infinite dimensional systems.

Reduces the problem to verifying basic properties like existence and tightness.

Provides a unified framework applicable to various models.

Abstract

The large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation. The construction and justification of the approximations can be onerous, especially in the case where the process state is infinite dimensional. In this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of Brownian noise. The approach is based on a variational representation for functionals of Brownian motion. Proofs of large deviations properties are reduced to demonstrating basic qualitative properties (existence, uniqueness and tightness) of certain perturbations of the original process.