Concentration bounds for stochastic systems with singular kernels

TL;DR

This paper establishes a link between entropic propagation of chaos and exponential concentration bounds for empirical measures in singular stochastic particle systems, providing a method to derive concentration bounds from controlled chaos assumptions.

Contribution

It introduces a variational upper bound approach connecting propagation of chaos with concentration inequalities for singular stochastic systems.

Findings

Proves exponential concentration bounds for empirical measures

Connects entropic chaos with concentration inequalities

Adapts existing methods to singular interaction systems

Abstract

This note is concerned with weakly interacting stochastic particle systems with possibly singular pairwise interactions. In this setting, we observe a connection between entropic propagation of chaos and exponential concentration bounds for the empirical measure of the system. In particular, we establish a variational upper bound for the probability of a certain rare event, and then use this upper bound to show that "controlled" entropic propagation of chaos implies an exponential concentration bound for the empirical measure. This connection allows us to infer concentration bounds for a class of singular stochastic systems through a simple adaptation of the arguments developed in Jabin and Wang (2018).