The Difference Between a Discrete and Continuous Harmonic Measure

Jianping Jiang; Tom Kennedy

doi:10.1007/s10959-016-0695-3

The Difference Between a Discrete and Continuous Harmonic Measure

Jianping Jiang, Tom Kennedy

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius h. For a simply connected domain D in the plane, let ω_h(0 , · ; D) be the discrete harmonic measure at 0 ∈ D associated with this random walk, and ω(0 , · ; D) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function σ_D(z) on ∂D such that for functions g which are in C² ⁺ ^α(∂D) for some α> 0 we have (Formula Presented.)We give an explicit formula for σ_D in terms of the conformal map from D to the unit disk. The proof relies on some fine approximations of the potential kernel and Green’s function of the random walk by their continuous counterparts, which may be of independent interest.

Original language	English (US)
Pages (from-to)	1424-1444
Number of pages	21
Journal	Journal of Theoretical Probability
Volume	30
Issue number	4
DOIs	https://doi.org/10.1007/s10959-016-0695-3
State	Published - Dec 1 2017

Keywords

Brownian motion
Dirichlet problem
Harmonic measure
Random walk

ASJC Scopus subject areas

Statistics and Probability
General Mathematics
Statistics, Probability and Uncertainty

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10.1007/s10959-016-0695-3

Cite this

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abstract = "We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius h. For a simply connected domain D in the plane, let ωh(0 , · ; D) be the discrete harmonic measure at 0 ∈ D associated with this random walk, and ω(0 , · ; D) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function σD(z) on ∂D such that for functions g which are in C2 + α(∂D) for some α> 0 we have (Formula Presented.)We give an explicit formula for σD in terms of the conformal map from D to the unit disk. The proof relies on some fine approximations of the potential kernel and Green{\textquoteright}s function of the random walk by their continuous counterparts, which may be of independent interest.",

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AB - We consider a discrete-time, continuous-state random walk with steps uniformly distributed in a disk of radius h. For a simply connected domain D in the plane, let ωh(0 , · ; D) be the discrete harmonic measure at 0 ∈ D associated with this random walk, and ω(0 , · ; D) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function σD(z) on ∂D such that for functions g which are in C2 + α(∂D) for some α> 0 we have (Formula Presented.)We give an explicit formula for σD in terms of the conformal map from D to the unit disk. The proof relies on some fine approximations of the potential kernel and Green’s function of the random walk by their continuous counterparts, which may be of independent interest.

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The Difference Between a Discrete and Continuous Harmonic Measure

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Keywords

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