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

TL;DR

This paper introduces a new, more transparent method for defining the self-intersection local time of planar Brownian motion by approximating it with simple random walks and using a discrete Tanaka--Rosen--Yor formula.

Contribution

It provides an elementary, almost sure approximation of planar Brownian self-intersection local time via simple random walks and a discrete Tanaka--Rosen--Yor formula.

Findings

Almost sure convergence of local averages of random walk self-intersection local times

Discrete Tanaka--Rosen--Yor formula approximates the continuous version

New approach is more transparent and elementary

Abstract

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