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

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

2010-08-05

arxiv.org/abs/1008.1006v2

Mathematics

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36 pages. A new part on renormalized self-intersection local time has been added and several inaccuracies have been corrected.

Scientific paper

10.1007/s10959-011-0351-x

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.

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