Continuum limit for some growth models

We derive a Hamilton–Jacobi equation for the macroscopic evolution of a class of growth models. For the definition of our growth models, we need a uniformly positive bounded continuous function λ : Rd×R→R which is uniformly Lipshitz in its last argument, and a nonnegative function v : Zd→Z+ with v(0)=0. The space of configurations Γ consist of functions h : Zd→Z such that h(i)−h(j)⩽v(i−j), for every pair of sites i,j∈Zd. We then take a sequence of independent Poisson clocks (p(i,k,·): (i,k)∈Zd×Z) of rates (λ(εi,εk): (i,k)∈Zd×Z). Initially, we start with a possibly random h∈Γ. The function h increases at site i∈Zd by one unit when the clock at site (i,h(i)+1) rings provided that h after the increase is still in Γ. In this way we have a process h(i,t) that after a rescaling uε(x,t)=εh([x/ε],tε) is expected to converge to a function u(x,t) that solves a Hamilton–Jacobi equation of the form ut+λ(x,u)H(ux)=0. We establish this provided that either λ is identically a constant or the set Γ can be described by some local constraints on the configuration h. When the Hamiltonian Ĥ(x,u,p)=λ(x,u)H(p) is not monotone in the u-variable, no uniqueness result is known for the solutions. Our method of derivation leads to a variational expression for the solutions that seems to be new. This variational expression offers a physically relevant candidate for a solution even if the uniqueness for the viscosity solutions fails.