| Abstract: |
| \begin{document}
\begin{abstract}
${\sf RCD}^*(K,N)$-spaces is a metric measure space generalizing Riemmanian manifolds with
lower Ricci bound $K\in \mathbb{R}$ and an upper bound $N\in[1,+\infty[$.
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 ${\sf RCD}^*(K,N)$-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.
\end{abstract}
\end{document} |
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