Brownian measures on Jordan-Virasoro curves associated to the Weil-Petersson metric
by Hélène Airault, Paul Malliavin and Anton Thalmaier
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 the complex plane requires probabilistic arguments well beyond the classical
theory of conformal welding, due to the lacking quasi-symmetry of canonical
Brownian motion on Diff(S¹). 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.
Journal of Functional Analysis 259 (2010) 3037-3079
The paper is available here:
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