Continuum limit for some growth models II
成果类型：
Article
署名作者：
Rezakhanlou, F
署名单位：
University of California System; University of California Berkeley
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1015345605
发表日期：
2001
页码：
1329-1372
关键词：
摘要：
We continue our investigations on a class of growth models introduced in a previous paper. Given a nonnegative function upsilon: Z(d) --> Z with upsilon(0) = 0, we define the space of configurations F to consist of functions h: Zd Z such that h(i)-h(j) < upsilon(i-j) for all i, j is an element of Z(d). We then take two sequences of independent Poisson clocks (p(+/-) (i, t): i is an element of Z(d)) of rates lambda(+/-). We start with a possibly random configuration h is an element of Gamma. 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 Gamma. Otherwise the change in h is suppressed. In this way we have a process h(i, t) that after a resealing u(epsilon)(x, t) = epsilonh([(x)/(epsilon)], (t)/(epsilon)) 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 the dimension is I 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 the processes u(epsilon) is a process u that satisfies a Hamilton-Jacobi equation for a suitable (possibly random) Hamiltonian H.