The global geometry of stochastic Lœwner evolutions

Abstract

In this article we develop a concise description of the global geometry which is underlying the universal construction of all possible generalised Stochastic Lœwner Evolutions. The main ingredient is the Universal Grassmannian of Sato–Segal–Wilson. We illustrate the situation in the case of univalent functions defined on the unit disc and the classical Schramm–Lœwner stochastic differential equation. In particular we show how the Virasoro algebra acts on probability measures. This approach provides the natural connection with Conformal Field Theory and Integrable Systems.