## EVENT DETAILS AND ABSTRACT

**Probability Seminar**
**Title:** New Laplacian comparison theorem and its applications to diffusion processes on Riemannian manifolds

**Speaker:** Kazuhiro Kuwae

**Speaker Info:** Fukuoka University

**Brief Description:**

**Special Note**:

**Abstract:**

Let $L= \Delta- \nabla \phi \cdot \nabla$ be a symmetric diffusion operator with an
invariant measure $\mu( dx )=e^{-\phi(x)} m( dx )$ on a complete non-compact smooth Riemannian manifold $(M,g)$ with its volume element $m$, and $\phi \in C^2(M)$ a potential function. In this talk, we show a Laplacian comparison theorem on weighted complete Riemannian manifolds with $\textrm{CD}(K, m)$-condition for $m \leq 1$ and a continuous function $K$. As consequences, we give the optimal conditions on $m$-Bakry-Emery Ricci tensor for $m\leq1$ such that the (weighted) Myers' theorem, Bishop-Gromov volume comparison theorem, Ambrose-Myers' theorem, and the Cheeger-Gromoll type splitting theorem, stochastic completeness and Feller property of $L$-diffusion processes hold on weighted complete Riemannian manifolds. This is a joint work with Xiang-Dong Li (Chinese Academy of Sciences).

**Date:** Tuesday, February 25, 2020

**Time:** 4:10PM

**Where:** Lunt 107

**Contact Person:** Elton Hsu

**Contact email:** ehsu@math.northwestern.edu

**Contact Phone:**

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