### Abstract

A simple extension is given of the well-known conformal invariance of harmonic measure in the plane. This equivalence depends on the interpretation of harmonic measure as an exit distribution of planar Brownian motion, and extends conformal invariance to analytic functions which are not injective, as well as allowing for stopping times more general than exit times. This generalization allow considerations of homotopy and reflection to be applied in order to compute new expressions for exit distributions of various domains, as well as the distribution of Brownian motion at certain other stopping times. An application of these methods is the derivation of a number of infinite sum identities, including the Leibniz formula for π and the values of the Riemann ξ function at even integers.

Original language | English |
---|---|

Pages (from-to) | 597-616 |

Number of pages | 20 |

Journal | Annales Academiae Scientiarum Fennicae Mathematica |

Volume | 43 |

DOIs | |

Publication status | Published - 1 Jan 2018 |

### Keywords

- Analytic functions
- Exit distribution
- Harmonic measure
- Planar Brownian motion

### Cite this

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**On the distribution of planar Brownian motion at stopping times.** / Markowsky, Greg.

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

TY - JOUR

T1 - On the distribution of planar Brownian motion at stopping times

AU - Markowsky, Greg

PY - 2018/1/1

Y1 - 2018/1/1

N2 - A simple extension is given of the well-known conformal invariance of harmonic measure in the plane. This equivalence depends on the interpretation of harmonic measure as an exit distribution of planar Brownian motion, and extends conformal invariance to analytic functions which are not injective, as well as allowing for stopping times more general than exit times. This generalization allow considerations of homotopy and reflection to be applied in order to compute new expressions for exit distributions of various domains, as well as the distribution of Brownian motion at certain other stopping times. An application of these methods is the derivation of a number of infinite sum identities, including the Leibniz formula for π and the values of the Riemann ξ function at even integers.

AB - A simple extension is given of the well-known conformal invariance of harmonic measure in the plane. This equivalence depends on the interpretation of harmonic measure as an exit distribution of planar Brownian motion, and extends conformal invariance to analytic functions which are not injective, as well as allowing for stopping times more general than exit times. This generalization allow considerations of homotopy and reflection to be applied in order to compute new expressions for exit distributions of various domains, as well as the distribution of Brownian motion at certain other stopping times. An application of these methods is the derivation of a number of infinite sum identities, including the Leibniz formula for π and the values of the Riemann ξ function at even integers.

KW - Analytic functions

KW - Exit distribution

KW - Harmonic measure

KW - Planar Brownian motion

UR - http://www.scopus.com/inward/record.url?scp=85055649559&partnerID=8YFLogxK

U2 - 10.5186/AASFM.2018.4338

DO - 10.5186/AASFM.2018.4338

M3 - Article

AN - SCOPUS:85055649559

VL - 43

SP - 597

EP - 616

JO - Annales Academiae Scientiarum Fennicae Mathematica

JF - Annales Academiae Scientiarum Fennicae Mathematica

SN - 1239-629X

ER -