Description:

Title: Stochastic Completeness of the Teichmuller Space

Abstract: The Teichmuller space T_g over a smooth oriented compact surface provides a parametrization of all Riemann surfaces of a fixed genus g. It has a natural complex structure of dimension 3g-3. Among several natural metrics on T_g, the most interesting one from analytic point of view is the Weil-Petersson metric, under which T_g becomes a Kaehler manifold of negative curvature diffeomorphic to a simply connected domain in the complex euclidean space. Our general goal is to study Brownian motion on T_g, more specifically how the geometry of Riemann surfaces change as it moves along a typical Brownian path, thus hopefully shedding lights on the structure of the Martin boundary of the Teichmuller space.  As a preliminary step, we show that T_g is stochastically complete, i.e., Brownian motion does not blow up in finite time (or it has infinite lifetime). This result provides a natural example of a conservative Riemannian Brownian motion on a simply connected manifold of negative curvature. The proof is based on the fact that the Teichmuller space projects via the mapping class group to the moduli space, which is an orbifold of finite volume and whose singular set is so small (more precisely, capacity zero) that with probability one Brownian motion does not meet.