Pathwise large deviations for white noise chaos expansions

TL;DR

This paper establishes large deviations principles for a family of white noise chaos expansions in continuous function spaces, extending previous results and providing new insights into pathwise deviations.

Contribution

It provides sufficient conditions for large deviations in white noise chaos expansions, addressing an open problem and introducing a novel perspective on pathwise large deviations.

Findings

Established large deviations principles for white noise chaos expansions.

Extended results to the Brownian motion case previously left open.

Introduced a new approach to analyze pathwise deviations using chaos representations.

Abstract

We consider a family of continuous processes $\{X^\varepsilon\}_{\varepsilon>0}$ which are measurable with respect to a white noise measure, take values in the space of continuous functions $C([0,1]^d:\mathbb{R})$, and have the Wiener chaos expansion \[ X^\varepsilon = \sum_{n=0}^{\infty} \varepsilon^n I_n \big(f_n^{\varepsilon} \big). \] We provide sufficient conditions for the large deviations principle of $\{X^\varepsilon\}_{\varepsilon>0}$ to hold in $C([0,1]^d:\mathbb{R})$, thereby refreshing a problem left open by P\'erez-Abreu (1993) in the Brownian motion case. The proof is based on the weak convergence approach to large deviations: it involves demonstrating the convergence in distribution of certain perturbations of the original process, and thus the main difficulties lie in analysing and controlling the perturbed multiple stochastic integrals. Moreover, adopting this representation offers a new perspective on pathwise large deviations and induces a variety of applications thereof.