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Continuum Limit for Some Growth Models II

We continue our investigations on a class of growth models introduced in a previous paper. Given a nonnegative function v: Z d → Z with v(0) = 0, we define the space of configurations Γ to consist of functions h: Z d → Z such that h(i)-h(j) ≤ v(i-j) for all i, j ∈ Z d . We then take two sequences of independent Poisson clocks (p ± (i, t): i e Z d ) of rates λ ± , We start with a possibly random configuration h ∈ Γ. The function h increases (respectively, decreases) by one unit at site i, when the clock p + (i, . ) [respectively, p - (i,.)] rings and the resulting configuration is still in Γ. Otherwise the change in h is suppressed. In this way we have a process h(i, t) that after a rescaling u e (x, t) = eh([x-e], t-e) is expected to converge to a function u(x, t) that solves a Hamilton-Jacobi equation of the form u t + H(u x ) = 0. We established this when A- or λ + = 0 in the previous paper, employing a strong monotonicity property of the process h(i, t). Such property is no longer available when both λ + , A- are nonzero. In this paper we initiate a new approach to treat the problem when the dimension is 1 and the set r can be described by local constraints on the configuration h. In higher dimensions, we can only show that any limit point of the processes u e is a process u that satisfies a Hamilton-Jacobi equation for a suitable (possibly random) Hamiltonian H.