Projecting the distribution of planar Browian motion at a stopping time through an analytic function

TL;DR

This paper introduces a method to determine the distribution of planar Brownian motion at specific stopping times using analytic functions, expanding conformal invariance concepts and providing new proofs of classical identities.

Contribution

It generalizes conformal invariance of harmonic measure to derive distributions at various stopping times, including non-exit times, and offers novel proofs of mathematical identities.

Findings

Derived distributions for Brownian motion at diverse stopping times.

Extended conformal invariance principles to broader classes of stopping times.

Provided new proofs of Euler's Basel sum and Leibniz's formula for π.

Abstract

A method is given of deriving the distribution of planar Brownian motion evaluated at certain stopping times using analytic functions. This method relies upon a generalization of the standard conformal invariance of harmonic measure. A number of examples are given, including several in which the stopping time in question is not the exit time of a domain. It is also shown how appropriate choices of domains and stopping times can lead to new proofs of identities, including Euler's Basel sum and a generalization of Leibniz's formula for $\pi$.