Hydrodynamics of the weakly asymmetric normalized binary contact path process

Abstract

We are concerned with the weakly asymmetric normalized binary contact path process. The symmetric version was introduced in Griffeath (1983). The model describes the spread of an infectious disease on the lattice Zd. The configuration at each site x∈Zd takes value in [0,∞). Site x is recovered at rate 1, and is infected by its neighbor x±ei at rate λ∓λ1∕N, where {ei}1≤i≤d is the canonical basis of the d-dimensional lattice, λ,λ1 are non-negative constants and N is the scaling parameter. When the infection occurs, the seriousness of the disease at site x is added with that of y. When there is neither recovery nor infection occurring during some time interval, the value at site x evolves according to some ODE. We prove that for d≥3 and large value λ, the empirical measure of the process, under diffusive scaling, converges in probability to a deterministic measure whose density is the unique weak solution to a linear parabolic equation, while for small λ and in all dimensions, the limit is zero. We also conjecture that the fluctuations field in the former case is driven by a generalized Ornstein–Uhlenbeck process, while a rigorous proof is absent. The main difficulty in proving the hydrodynamics is to prove the absolute continuity of the limiting path, where the theory of linear systems introduced in Liggett (1985) is utilized.