Shy and fixed distance couplings on Riemanian manifolds

Stochastics Seminar
Thursday, September 5, 2013 - 15:05
1 hour (actually 50 minutes)
We show that on any Riemannian manifold with the Ricci curvature non-negative we can construct a coupling of two Brownian motions which are staying fixed distance for all times.  We show a more general version of this for the case of Ricci bounded from below uniformly by a constant k.   In the terminology of Burdzy, Kendall and others, a shy coupling is a coupling in which the Brownian motions do not couple in finite time with positive probability.   What we construct here is a strong version of shy couplings on Riemannian manifolds.   On the other hand, this can be put in contrast with some results of von Renesse and K. T. Sturm which give a characterization of the lower bound on the Ricci curvature in terms of couplings of Brownian motions and our construction optimizes this choice in a way which will be explained.  This is joint work with Mihai N. Pascu.