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July 2005 Barycenters of measures transported by stochastic flows
Marc Arnaudon, Xue-Mei Li
Ann. Probab. 33(4): 1509-1543 (July 2005). DOI: 10.1214/009117905000000071


We investigate the evolution of barycenters of masses transported by stochastic flows. The state spaces under consideration are smooth affine manifolds with certain convexity structure. Under suitable conditions on the flow and on the initial measure, the barycenter {Zt} is shown to be a semimartingale and is described by a stochastic differential equation. For the hyperbolic space the barycenter of two independent Brownian particles is a martingale and its conditional law converges to that of a Brownian motion on the limiting geodesic. On the other hand for a large family of discrete measures on suitable Cartan–Hadamard manifolds, the barycenter of the measure carried by an unstable Brownian flow converges to the Busemann barycenter of the limiting measure.


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Marc Arnaudon. Xue-Mei Li. "Barycenters of measures transported by stochastic flows." Ann. Probab. 33 (4) 1509 - 1543, July 2005.


Published: July 2005
First available in Project Euclid: 1 July 2005

zbMATH: 1077.60039
MathSciNet: MR2150197
Digital Object Identifier: 10.1214/009117905000000071

Primary: 60G60
Secondary: 60F05 , 60F15 , 60G44 , 60G57 , 60H10 , 60J65

Keywords: Brownian motion , Busemann barycenter , convex geometry , Exponential barycenter , Hyperbolic space , manifold with connection , stochastic flow

Rights: Copyright © 2005 Institute of Mathematical Statistics


Vol.33 • No. 4 • July 2005
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