On the planar Brownian Green's function for stopping times

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:455

Article

Markowsky, Greg¹ 

[1] Monash Univ, Clayton, Vic 3800, Australia

关键词: Planar Brownian motion;    Green's function;    Analytic function theory;    Riemann mapping theorem;    Infinite products;   

DOI  :  10.1016/j.jmaa.2017.06.013

来源: Elsevier

【 摘 要 】

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. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free