Continuum Limit for Some Growth Models II

Abstract

We continue our investigations on a class of growth models introduced in a previous paper. Given a nonnegative function $v: \mathbb{Z}^d \to \mathbb{Z}$ with $v(0) = 0$, we define the space of configurations $\Gamma$ to consist of functions $h: \mathbb{Z}^d \to \mathbb{Z}$ such that $h(i) - h(j) \leq v(i - j)$ for all i$i, j \epsilon \mathbb{Z}^d$. We then take two sequances of independent Poisson clocks $(p^{\pm}(i, t): i \epsilon \mathbb{Z}^d)$ of rates $\lambda^{\pm}$. We start with a possibly random configuration $h \epsilon \Gamma$. The function $h$ increases (respectively, decreases) by one unit at site $i$, when the clock $p^{+}(i, \cdot)$ [respectively, $p^{-}(i, \cdot)$] rings and the resulting configuration is still in $\Gamma$. Otherwise the change in h is suppressed. In this way we have a process $h(i, t)$ that after a rescaling $u^{\varepsilon}(x, t) = \varepsilonh([\frac{x}{\varepsilon}], \frac{t}{\varepsilon})$ 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 $\lambda^{-}$ or $\lambda^{+} = 0$ in the previous paper, employing a strong monotonicity property of the process $h(i, t)$. Such property is no longer available when both $\lambda^{+}, \lambda^{-}$ are nonzero. In this paper we initiate a new approach to treat the problem when th dimension is 1 and the set $\Gamma$ can be described by local constraints on the configuration $h$. In higher dimensions, we can only show that any limit point of th processes $u^{varepsilon}$ is a process $u$ that satisfies a Hamilton-Jacobi equation for a suitable (possibly random) Hamiltonian $H$.