ERGODIC-THEOREMS FOR INFINITE SYSTEMS OF LOCALLY INTERACTING DIFFUSIONS

• 成果类型: Article
• 署名作者: COX, JT; GREVEN, A
• 署名单位: University of Gottingen
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988732
• 发表日期: 1994
• 页码: 833-853
• 关键词: differential-equations; PHASE-TRANSITION; time

摘要

Let x(t) = {x(i)(t), i is-an-element-to Z(d)} be the solution of the system of stochastic differential equations dx(i)(t) = (SIGMA(j is-an-element-of Z(d) a (i, j)x(j)(t) - x(i)(t)) dt + square-root 2g(x(i)(t)) dw(i)(t), i is-an-element-of Z(d). Here g: [0, 1] - R+ satisfies g > 0 on (0, 1), g(0) = g(1) = 0, g is Lipschitz, a(i,j) is an irreducible random walk kernel on Z(d) and {w(i)(t), i is-an-element-of Z(d)) is a family of standard, independent Brownian motions on R; x(t) is a Markov process on X = [0, 1]Z(d). This class of processes was studied by Notohara and Shiga; the special case g(v) = v(1 - v) has been studied extensively by Shiga. We show that the long term behavior of x(t) depends only on a(i,j) = (a(i,j) + a(j,i))/2 and is universal for the entire class of g considered. If a(i,j) is transient then there exists a family {nu(theta), theta is-an-element-of [0, 1]} of extremal, translation invariant equilibria. Each nu(theta) is mixing and has density theta = integral x0 dnu(theta). If a(i,j), is recurrent, then the set of extremal translation invariant equilibria consist of the point masses {delta0, delta1}. The process starting in a translation invariant, shift ergodic measure mu on X with integral x0 dmu = theta converges weakly as t --> infinity to nu(theta) if a(i,j) is transient, and to (1 - theta)delta + thetadelta, if a(i,j) is recurrent. (Our results in the recurrent case remove a mild assumption on g imposed by Notohara and Shiga.) For the case a(i,j) transient we use methods developed for infinite particle systems by Liggett and Spitzer. For the case a(i,j), recurrent we use a duality comparison argument.