# Correlation function methods for a system of annihilating Brownian particles

## TL;DR

This paper discusses the correlation function method as a unified approach to deriving hydrodynamic and fluctuation limits for reaction-diffusion particle systems, focusing on a specific nonlinear PDE model.

## Contribution

It provides an expository overview of the correlation function method applied to reaction-diffusion systems, simplifying previous proofs by using reflected Brownian motion.

## Key findings

- Unified approach for hydrodynamic and fluctuation limits
- Simplified proof framework using reflected Brownian motion
- Application to a nonlinear reaction-diffusion PDE

## Abstract

In this expository note we highlight the correlation function method as a unified approach in proving both hydrodynamic limits and fluctuation limits for reaction diffusion particle systems. For simplicity we focus on the case when the hydrodynamic limit is $\partial_t u=\frac{1}{2}\Delta u -u^2$, one of the simplest nonlinear reaction-diffusion equations. The outline of the proof follows from Chapter 4 of De Masi and Presutti [7] but to simplify the presentation, we consider reflected Brownian motion instead of reflected random walks. We also briefly mention the key ideas in proving the fluctuation result.