The Annals of Probability

Barycenters of measures transported by stochastic flows

Marc Arnaudon and Xue-Mei Li

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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.

Article information

Ann. Probab., Volume 33, Number 4 (2005), 1509-1543.

First available in Project Euclid: 1 July 2005

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Zentralblatt MATH identifier

Primary: 60G60: Random fields
Secondary: 60G57: Random measures 60H10: Stochastic ordinary differential equations [See also 34F05] 60J65: Brownian motion [See also 58J65] 60G44: Martingales with continuous parameter 60F05: Central limit and other weak theorems 60F15: Strong theorems

Exponential barycenter Busemann barycenter stochastic flow manifold with connection convex geometry hyperbolic space Brownian motion


Arnaudon, Marc; Li, Xue-Mei. Barycenters of measures transported by stochastic flows. Ann. Probab. 33 (2005), no. 4, 1509--1543. doi:10.1214/009117905000000071.

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