Self-intersection local time of planar Brownian motion based on a strong approximation by random walks

摘要

The main purpose of this work is to define planar self-intersection localtime by an alternative approach which is based on an almost sure pathwiseapproximation of planar Brownian motion by simple, symmetric random walks. As aresult, Brownian self-intersection local time is obtained as an almost surelimit of local averages of simple random walk self-intersection local times. Animportant tool is a discrete version of the Tanaka--Rosen--Yor formula; thecontinuous version of the formula is obtained as an almost sure limit of thediscrete version. The author hopes that this approach to self-intersectionlocal time is more transparent and elementary than other existing ones.