Large Deviations of the Front in a One

We investigate the probabilities of large deviations for the position of the front in a stochastic model of the reaction X + Y → 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.