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

Jean Bérard (1) , Alejandro Ramírez (2)

1 ICJ - Institut Camille Jordan
2 Facultad de Matemáticas [Santiago de Chile]

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.

Domaines: Probabilités [math.PR]

Jean Bérard, Alejandro Ramírez. Large Deviations of the Front in a one dimensional model of $X+Y \to 2X$. Annals of Probability, 2010, 38 (3), pp.955-1018. ⟨hal-00296713⟩