### 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 language | English (US) |
---|---|

Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Journal of Theoretical Probability |

DOIs | |

State | Accepted/In press - Jun 7 2016 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

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

### Cite this

*Journal of Theoretical Probability*, 1-21. https://doi.org/10.1007/s10959-016-0695-3

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

Research output: Contribution to journal › Article

}

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 -