Brownian measures on Jordan–Virasoro curves associated to the Weil–Petersson metric

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• 作者：Hé ; lè ; ne Airault ; Paul Malliavin ; Anton Thalmaier
• 关键词：Jordan curves ; Conformal welding ; Weil– ; Petersson metric ; Douady– ; Earle extension ; Group of diffeomorphisms ; Kä ; hler Brownian motion ; Shape representation
• 刊名：Journal of Functional Analysis
• 出版年：2010
• 出版时间：15 December 2010
• 年：2010
• 卷：259
• 期：12
• 页码：3037-3079
• 全文大小：297 K

文摘

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 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.