随机流输运的测度重心
Barycenters of measures transported by stochastic flows
摘要：
本文研究了由随机流输运的质量重心的演化。所考虑的状态空间是具有一定凸性结构的光滑仿射流形。在适当的流和初始测度条件下，重心{Z_t}是一个半鞅，并由一个随机微分方程描述.在双曲空间中，两个独立布朗粒子的重心是鞅，其条件律收敛于极限测地线上的布朗运动的条件律。另一方面，对于适当的Cartan-Hadamard流形上的一大族离散测度，不稳定布朗流所携带的测度的重心收敛到极限测度的Busemann重心。
英文摘要：
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 {Z_t} 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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