The Difference Between a Discrete and Continuous Harmonic Measure

Jianping Jiang, Thomas G Kennedy

Research output: Contribution to journalArticle

1 Citation (Scopus)

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 (Formula presented.) be the discrete harmonic measure at (Formula presented.) associated with this random walk, and (Formula presented.) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function (Formula presented.) on (Formula presented.) such that for functions g which are in (Formula presented.) for some (Formula presented.) we have (Formula presented.)We give an explicit formula for (Formula presented.) 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 languageEnglish (US)
Pages (from-to)1-21
Number of pages21
JournalJournal of Theoretical Probability
DOIs
StateAccepted/In press - Jun 7 2016

Fingerprint

Harmonic Measure
Random walk
Conformal Map
G-function
Potential Function
Kernel Function
Unit Disk
Green's function
Explicit Formula
Continuous Function
Discrete-time
Radius

Keywords

  • Brownian motion
  • Dirichlet problem
  • Harmonic measure
  • Random walk

ASJC Scopus subject areas

  • Mathematics(all)
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

The Difference Between a Discrete and Continuous Harmonic Measure. / Jiang, Jianping; Kennedy, Thomas G.

In: Journal of Theoretical Probability, 07.06.2016, p. 1-21.

Research output: Contribution to journalArticle

@article{a5d7f4e00e9e4cb49eff99778342c787,
title = "The Difference Between a Discrete and Continuous Harmonic Measure",
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 (Formula presented.) be the discrete harmonic measure at (Formula presented.) associated with this random walk, and (Formula presented.) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function (Formula presented.) on (Formula presented.) such that for functions g which are in (Formula presented.) for some (Formula presented.) we have (Formula presented.)We give an explicit formula for (Formula presented.) 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.",
keywords = "Brownian motion, Dirichlet problem, Harmonic measure, Random walk",
author = "Jianping Jiang and Kennedy, {Thomas G}",
year = "2016",
month = "6",
day = "7",
doi = "10.1007/s10959-016-0695-3",
language = "English (US)",
pages = "1--21",
journal = "Journal of Theoretical Probability",
issn = "0894-9840",
publisher = "Springer New York",

}

TY - JOUR

T1 - The Difference Between a Discrete and Continuous Harmonic Measure

AU - Jiang, Jianping

AU - Kennedy, Thomas G

PY - 2016/6/7

Y1 - 2016/6/7

N2 - 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 (Formula presented.) be the discrete harmonic measure at (Formula presented.) associated with this random walk, and (Formula presented.) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function (Formula presented.) on (Formula presented.) such that for functions g which are in (Formula presented.) for some (Formula presented.) we have (Formula presented.)We give an explicit formula for (Formula presented.) 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.

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 (Formula presented.) be the discrete harmonic measure at (Formula presented.) associated with this random walk, and (Formula presented.) be the (continuous) harmonic measure at 0. For domains D with analytic boundary, we prove there is a bounded continuous function (Formula presented.) on (Formula presented.) such that for functions g which are in (Formula presented.) for some (Formula presented.) we have (Formula presented.)We give an explicit formula for (Formula presented.) 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.

KW - Brownian motion

KW - Dirichlet problem

KW - Harmonic measure

KW - Random walk

UR - http://www.scopus.com/inward/record.url?scp=84976293609&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976293609&partnerID=8YFLogxK

U2 - 10.1007/s10959-016-0695-3

DO - 10.1007/s10959-016-0695-3

M3 - Article

AN - SCOPUS:84976293609

SP - 1

EP - 21

JO - Journal of Theoretical Probability

JF - Journal of Theoretical Probability

SN - 0894-9840

ER -