Abstract:

RCD*(K,N)-spaces is a metric measure space generalizing Riemmanian manifolds with lower Ricci bound K(in R) and an upper bound N being greater than 1 and finite.

This class of spaces also contains the class of N-dimenasional Alexandrov spaces, which was proved by Petrunin and Zhang-Zhu, and also contains the class of weighted Riemannian manifolds with Witten Laplacian and lower bound K of $N$-Bakry-Emery Ricci tensor. I will talk on new stochastic expression of radial process under the law for all starting point including the reference point appeared in the radial function provided the reference point fulfills a regularity condition depending on the geometric structure of the RCD*(K,N)-space.

The expression of radial process is completely different from Kendall’s expression(1987) including the local time on cut-locus without lower Ricci bound in the framework of Riemannian manifold. Our expression of radial process does not contain the local time on cut-locus. Instead of it, we extract a positive continuous additive functional, which can be thought of continuous additive functionals corresponding to the difference of Laplacians of radial functions between on the given space and on the model space.

This is a joint work with Kazumasa Kuwada in Tohoku University.