Large Deviations for Brownian Particle Systems with Killing

TL;DR

This paper establishes a large deviation principle for systems of Brownian particles with killing, providing insights into their probabilistic behavior and a new variational representation for related expectations.

Contribution

It introduces a large deviation principle for particle systems with killing and a novel variational representation for expectations involving Brownian motions and i.i.d. variables.

Findings

LDP for sub-probability measure-valued processes

A new variational representation for expectations of Brownian functionals

Application of weak convergence methods in proof

Abstract

Particle approximations for certain nonlinear and nonlocal reaction-diffusion equations are studied using a system of Brownian motions with killing. The system is described by a collection of i.i.d. Brownian particles where each particle is killed independently at a rate determined by the empirical sub-probability measure of the states of the particles alive. A large deviation principle (LDP) for such sub-probability measure-valued processes is established. Along the way a convenient variational representation, which is of independent interest, for expectations of nonnegative functionals of Brownian motions together with an i.i.d. sequence of random variables is established. Proof of the LDP relies on this variational representation and weak convergence arguments.