Brownian bridges on manifolds

Brownian bridges are natural stochastic processes on a manifold conditioned on the starting and ending points given some running time. If we fixed the starting and ending points and let its running time go to 0, the path of the brownian bridge will behave like the geodesics and the law has some large deviation property. In this talk, I will first give a brief introduction of large deviations and the work by Pei Hsu in 1990, then introduce a similar problem but under the setting of a current research area called Liouville quantum gravity.