Stochastic Completeness of Brownian Motion on the Teichmuller Space

YMSC Probability Seminar

Organizers：

吴昊，杨帆，姜建平，顾陈琳

Speaker：

Elton Hsu 徐佩

Northwest University

Time：

Thur., 4:00-5:00 pm, Sept. 11, 2025

Venue：

C548, Shuangqing Complex Building A

Title：

Stochastic Completeness of Brownian Motion on the Teichmuller Space

Abstract:

The Teichmuller space T_g is a way to parametrize the space of complex structures of a compact surface of a fixed genus g. Under the Weil-Petersson metric It is negatively Kahler manifold of dimension 3g - 3 (for g greater than 1) diffeomorphic to a simply connected domain in the complex euclidean space. I propose to investigate an ideal boundary (e.g., the Martin boundary) of the Teichmuller space using the Riemannian Brownian motion on T_g under the Weil-Petersson metric. As a first step, in this talk, I show that the Riemannian Brownian motion is stochastically complete, i.e., it has an infinite lifetime (with probability one). Thus the Teichmuller space is a natural example of a geometrically incomplete, but stochastically complete Riemannian manifold. This approach also opens a way of investigating boundary properties of the Teichmuller space via its Martin compactification through the limiting behavior of the Riemannian Brownian motion as time goes to infinity.