Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks

• 成果类型: Article
• 署名作者: Chen, X
• 署名单位: University of Tennessee System; University of Tennessee Knoxville
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117904000000513
• 发表日期: 2004
• 页码: 3248-3300
• 关键词: large deviations; aleatory walks; sobolev; range

摘要

Let alpha([0, 1](P)) denote the intersection local time of p independent d-dimensional Brownian motions running up to the time 1. Under the conditions p(d - 2) < d and d greater than or equal to 2, we prove lim(t-->infinity) t(-1) logP{alpha([0, 1](P)) greater than or equal to t((d(p-1))/2)} = -gammaalpha(d, p) with the right-hand side being identified in terms of the the best constant of the Gagliardo-Nirenberg inequality. Within the scale of moderate deviations, we also establish the precise tail asymptotics for the intersection local time I-n = #{(k(1),...,k(P)) is an element of [1,n](P) ; S-1(k(1)) = ... = S-P(k(P))} run by the independent, symmetric, Zd-valued random walks S-1(n),....,S-P(n). Our results apply to the law of the iterated logarithm. Our approach is based on Feynman-Kac type large deviation, time exponentiation, moment computation and some technologies along the lines of probability in Banach space. As an interesting coproduct, we obtain the inequality [GRAPHICS] in the case of random walks.