Large Deviations of the Front in a one dimensional model of $X+Y \to 2X$

TL;DR

This paper establishes a large deviations principle for the front position in a stochastic reaction model on a lattice, providing insights into the probabilities of significant deviations from the expected front velocity.

Contribution

It proves a large deviations principle for the front in a reaction-diffusion lattice model and characterizes the rate function's zero set, revealing a gapless property similar to nonlinear diffusion equations.

Findings

Large deviations principle holds for the front position.

Zero set of the rate function is the interval [0,v].

Provides estimates for slowdown probability decay rates.

Abstract

We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction $X+Y \to 2X$ on the integer lattice in which $Y$ particles do not move while $X$ particles move as independent simple continuous time random walks of total jump rate $2$. For a wide class of initial conditions, we prove that a large deviations principle holds and we show that the zero set of the rate function is the interval $[0,v]$, where $v$ is the velocity of the front given by the law of large numbers. We also give more precise estimates for the rate of decay of the slowdown probabilities. Our results indicate a gapless property of the generator of the process as seen from the front, as it happens in the context of nonlinear diffusion equations describing the propagation of a pulled front into an unstable state.