On the planar Brownian Green's function for stopping times

TL;DR

This paper extends the conformal invariance of the planar Brownian Green's function to non-injective analytic functions, enabling new calculations for winding stopping times and providing a novel proof of the Riemann mapping theorem.

Contribution

It introduces a generalized conformal invariance concept for Green's functions, applicable to non-injective functions, and applies it to winding times and classical identities.

Findings

Calculated Green's function for winding stopping times

Provided a new proof of the Riemann mapping theorem

Derived identities for trigonometric functions

Abstract

It has been known for some time that the Green's function of a planar domain can be defined in terms of the exit time of Brownian motion, and this definition has been extended to stopping times more general than exit times. In this paper, we extend the notion of conformal invariance of Green's function to analytic functions which are not injective, and use this extension to calculate the Green's function for a stopping time defined by the winding of Brownian motion. These considerations lead to a new proof of the Riemann mapping theorem. We also show how this invariance can be used to deduce several identities, including the standard infinite product representations of several trigonometric functions.