Some new classes of exceptional times of linear Brownian motion
成果类型：
Article
署名作者：
Aspandiiarov, S; LeGall, JF
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/aop/1176987795
发表日期：
1995
页码：
1605-1626
关键词：
摘要：
We study certain classes of exceptional times of a linear Brownian motion (B-t, t greater than or equal to 0). In particular, we consider the set K- of all instants t is an element of [0, 1] such that the value B-t of the Brownian motion at time t is greater than its mean value over all intervals [s, t], s < t. We also study the subset K of K- of all instants t such that in addition B, is greater than the mean value of B over the intervals [t, s], t < s less than or equal to 1. We compute the Hausdorff dimension of K-, K and some other related sets of exceptional times. These results are closely related to a recent work of Sinai motivated by the analysis of solutions to the Burgers equation with random initial data. The proofs involve studying suitable approximations of the sets K- and K, and deriving precise estimates for the probability that a given time t belongs to these approximations. A delicate zero-one law argument is also needed to prove that the lower bounden the dimension of K holds with probability 1.
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