Brownian measures on Jordan-Virasoro curves associated to Weil-Petersson metric

In this paper existence of the Brownian measure on Jordan curves with respect to the Weil-Petersson metric is established. The step from Brownian motion on the diffeomorphism group of the circle to Brownian motion on Jordan curves in C requires probabilistic arguments well beyond the classical theory of conformal welding, due to the lacking quasi-symmetry of canonical Brownian motion on Diff(S 1 ). A new key step in our construction is the systematic use of a Kählerian diffusion on the space of Jordan curves for which the welding functional gives rise to conformal martingales, together with a Douady-Earle type conformal extension of vector fields on the circle to the disk