On the form of the large deviation rate function for the empirical measures of weakly interacting systems

TL;DR

This paper explores the form of the large deviation rate function for empirical measures in weakly interacting systems, extending known results beyond the case of tilted distributions to broader scenarios.

Contribution

It characterizes when the large deviation rate function can be expressed as relative entropy in weakly interacting systems beyond classical tilted measure cases.

Findings

Rate function expressed as relative entropy in broader scenarios

Extension of large deviation principles beyond tilted distributions

Conditions for large deviation principles in weakly interacting systems

Abstract

A basic result of large deviations theory is Sanov's theorem, which states that the sequence of empirical measures of independent and identically distributed samples satisfies the large deviation principle with rate function given by relative entropy with respect to the common distribution. Large deviation principles for the empirical measures are also known to hold for broad classes of weakly interacting systems. When the interaction through the empirical measure corresponds to an absolutely continuous change of measure, the rate function can be expressed as relative entropy of a distribution with respect to the law of the McKean-Vlasov limit with measure-variable frozen at that distribution. We discuss situations, beyond that of tilted distributions, in which a large deviation principle holds with rate function in relative entropy form.