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March 2008 The scaling limits of planar LERW in finitely connected domains
Dapeng Zhan
Ann. Probab. 36(2): 467-529 (March 2008). DOI: 10.1214/07-AOP342

Abstract

We define a family of stochastic Loewner evolution-type processes in finitely connected domains, which are called continuous LERW (loop-erased random walk). A continuous LERW describes a random curve in a finitely connected domain that starts from a prime end and ends at a certain target set, which could be an interior point, or a prime end, or a side arc. It is defined using the usual chordal Loewner equation with the driving function being $\sqrt{2}B(t)$ plus a drift term. The distributions of continuous LERW are conformally invariant. A continuous LERW preserves a family of local martingales, which are composed of generalized Poisson kernels, normalized by their behaviors near the target set. These local martingales resemble the discrete martingales preserved by the corresponding LERW on the discrete approximation of the domain. For all kinds of targets, if the domain satisfies certain boundary conditions, we use these martingales to prove that when the mesh of the discrete approximation is small enough, the continuous LERW and the corresponding discrete LERW can be coupled together, such that after suitable reparametrization, with probability close to 1, the two curves are uniformly close to each other.

Citation

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Dapeng Zhan. "The scaling limits of planar LERW in finitely connected domains." Ann. Probab. 36 (2) 467 - 529, March 2008. https://doi.org/10.1214/07-AOP342

Information

Published: March 2008
First available in Project Euclid: 29 February 2008

zbMATH: 1153.60057
MathSciNet: MR2393989
Digital Object Identifier: 10.1214/07-AOP342

Subjects:
Primary: 60G17, 82B41

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 2 • March 2008
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