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The Annals of Probability 2005, Vol. 33, No. 4, 15091543
 

Summary: The Annals of Probability
2005, Vol. 33, No. 4, 1509­1543
DOI 10.1214/009117905000000071
© Institute of Mathematical Statistics, 2005
BARYCENTERS OF MEASURES TRANSPORTED BY
STOCHASTIC FLOWS
BY MARC ARNAUDON AND XUE-MEI LI1
Université de Poitiers and Nottingham Trent University
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 Cartan­Hadamard manifolds, the barycenter of the
measure carried by an unstable Brownian flow converges to the Busemann
barycenter of the limiting measure.
1. Introduction. We consider the motion of a mass moving according to the

  

Source: Arnaudon, Marc - Département de mathématiques, Université de Poitiers
Li, Xue-Mei - Mathematics Institute, University of Warwick

 

Collections: Mathematics