ASYMPTOTIC EXPANSION OF THE INVARIANT MEASURE FOR BALLISTIC RANDOM WALK IN THE LOW DISORDER REGIME

• 成果类型: Article
• 署名作者: Campos, David; Ramirez, Alejandro F.
• 署名单位: Universidad Costa Rica; Pontificia Universidad Catolica de Chile
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1175
• 发表日期: 2017
• 页码: 4675-4699
• 关键词: random environment; large numbers; of-view; points; LAW

摘要

We consider a random walk in random environment in the low disorder regime on Z(d), that is, the probability that the random walk jumps from a site x to a nearest neighboring site x + e is given by p(e) + epsilon xi(x, e), where p(e) is deterministic, {{xi(x, e) : vertical bar e vertical bar(1) = 1} : x is an element of Z(d)} are i.i.d. and epsilon > 0 is a parameter, which is eventually chosen small enough. We establish an asymptotic expansion in epsilon for the invariant measure of the environmental process whenever a ballisticity condition is satisfied. As an application of our expansion, we derive a numerical expression up to first order in epsilon for the invariant measure of random perturbations of the simple symmetric random walk in dimensions d = 2.