Annales Academiæ Scientiarum Fennicæ

Mathematica

Volumen 43, 2018, 597-616

# ON THE DISTRIBUTION OF PLANAR
BROWNIAN MOTION AT STOPPING TIMES

## Greg Markowsky

Monash University

Victoria 3800, Australia;
gmarkowsky 'at' gmail.com

**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 ζ function at even integers.

**2010 Mathematics Subject Classification:**
Primary 60J65, 30A99.

**Key words:**
Planar Brownian motion, analytic functions, harmonic measure, exit distribution.

**Reference to this article:** G. Markowsky:
On the distribution of planar Brownian motion at stopping times.
Ann. Acad. Sci. Fenn. Math. 43 (2018), 597-616.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4338

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