Large deviations in the supremum norm for a reaction–diffusion system

Abstract

We present large deviations estimates in the supremum norm for a system of independent random walks superposed with a birth-and-death dynamics evolving on the discrete torus with N sites. The scaling limit considered is the so-called high density limit (see the survey in (In From Particle Systems to Partial Differential Equations (2014) 179–189 Springer) on the subject), where space, time and initial quantity of particles are rescaled. The associated rate functional here obtained is a semi-linearized version of the rate function of (Probability Theory and Related Fields 97 (1992) 339–361), which dealt with large deviations of exclusion processes superposed with birth-and-death dynamics. An important ingredient in the proof of large deviations consists in providing a limit of a suitable class of perturbations of the original process, which is precisely one of the main contributions of this work: a strategy to extend the original high density approach (as in Advances in Applied Probability 12 (1980) 367–379; Annals of Probability 19 (1991) 1440–1462; The Annals of Applied Probability 2 (1992) 131–141; Journal of Statistical Physics 149 (2012) 629–642; Annals of Probability 14 (1986) 173–193; Probability Theory and Related Fields 78 (1988) 11–37) to weakly asymmetric systems. Two cases are considered with respect to the initial quantity of particles, the power law and the (at least) exponential growth. In the first case, we present the lower bound only on a certain set of smooth profiles, while in the second case under some extra technical assumptions we provide a full large deviations principle.