On singularity as a function of time of a conditional distribution of an exit time
成果类型：
Article
署名作者：
Krylov, N. V.
署名单位：
University of Minnesota System; University of Minnesota Twin Cities
刊物名称：
PROBABILITY THEORY AND RELATED FIELDS
ISSN/ISSBN：
0178-8051
DOI：
10.1007/s00440-015-0639-3
发表日期：
2016
页码：
541-557
关键词：
摘要：
We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability distribution that the sum of two one-dimensional Brownian motions will exit from the unit interval before time t, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffusion of the Brownian motion is sufficiently large.
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