Canonical Brownian motion on the space of univalent functions and resolution of Beltrami equations by a continuity method along stochastic flows

Abstract

A.A. Kirillov has given a parametrization of the space U ∞ of univalent functions on the closed unit disk, which are C ∞ up to the boundary, by Diff ( S 1 ) / S 1 where Diff ( S 1 ) denotes the group of orientation preserving diffeomorphisms of the circle S 1 . In the same spirit, the space J ∞ of C ∞ Jordan curves in the complex plane can be parametrized by the double quotient SU ( 1 , 1 ) \\ Diff ( S 1 ) / SU ( 1 , 1 ) . As a consequence, J ∞ carries a canonical Riemannian metric. We construct a canonical Brownian motion on U ∞ . Classical technologies of the theory of univalent functions, like Beurling–Ahlfors extension, Loewner equation, Beltrami equation, developed in the context of Kunita's stochastic flows, are the tools for obtaining this result which should be seen as a first step to the construction of a canonical Brownian motion on J ∞ .