The Annals of Probability

The scaling limits of planar LERW in finitely connected domains

Dapeng Zhan

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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.

Article information

Source
Ann. Probab., Volume 36, Number 2 (2008), 467-529.

Dates
First available in Project Euclid: 29 February 2008

Permanent link to this document
https://projecteuclid.org/euclid.aop/1204306959

Digital Object Identifier
doi:10.1214/07-AOP342

Mathematical Reviews number (MathSciNet)
MR2393989

Zentralblatt MATH identifier
1153.60057

Subjects
Primary: 60G17: Sample path properties 82B41: Random walks, random surfaces, lattice animals, etc. [See also 60G50, 82C41]

Keywords
Loop-erased random walk scaling limit stochastic Loewner evolution

Citation

Zhan, Dapeng. The scaling limits of planar LERW in finitely connected domains. Ann. Probab. 36 (2008), no. 2, 467--529. doi:10.1214/07-AOP342. https://projecteuclid.org/euclid.aop/1204306959


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