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.
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Szabados, T. Self-intersection Local Time of Planar Brownian Motion Based on a Strong Approximation by Random Walks. J Theor Probab 25, 1081–1118 (2012). https://doi.org/10.1007/s10959-011-0351-x
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DOI: https://doi.org/10.1007/s10959-011-0351-x