The Global Geometry of Stochastic L{\oe}wner Evolutions

In this article we develop a concise description of the global geometry which is underlying the universal construction of all possible generalised Stochastic L{\oe}wner 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-L{\oe}wner 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.