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 which is uniformly Lipshitz in its last argument, and a nonnegative function with v(0)=0. The space of configurations [Gamma] consist of functions such that h(i)-h(j)[less-than-or-equals, slant]v(i-j), for every pair of sites . We then take a sequence of independent Poisson clocks of rates . Initially, we start with a possibly random h[set membership, variant][Gamma]. The function h increases at site by one unit when the clock at site (i,h(i)+1) rings provided that h after the increase is still in [Gamma]. In this way we have a process h(i,t) that after a rescaling is expected to converge to a function u(x,t) that solves a Hamilton-Jacobi equation of the form ut+[lambda](x,u)H(ux)=0. We establish this provided that either [lambda] is identically a constant or the set [Gamma] can be described by some local constraints on the configuration h. When the Hamiltonian 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.