The Difference Between a Discrete and Continuous Harmonic Measure

Jianping Jiang, Thomas G Kennedy

Research output: Contribution to journalArticle

1 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 (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

Keywords

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

ASJC Scopus subject areas

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

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