On the planar Brownian Green's function for stopping times

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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.

Original languageEnglish
Pages (from-to)1221–1233
Number of pages13
JournalJournal of Mathematical Analysis and Applications
Volume455
Issue number2
DOIs
Publication statusPublished - 2017

Keywords

  • Analytic function theory
  • Green's function
  • Infinite products
  • Planar Brownian motion
  • Riemann mapping theorem

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