The Difference Between a Discrete and Continuous Harmonic Measure

Jianping Jiang, Tom Kennedy

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)1424-1444
Number of pages21
JournalJournal of Theoretical Probability
Issue number4
StatePublished - Dec 1 2017


  • 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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