: Gradient estimates for Brownian bridges to submanifolds

Abstract: A diffusion process on a Riemannian manifold whose generator is one half of the Laplacian is called a Brownian motion. The mean local time of Brownian motion on a hypersurface will be considered, as will the situation in which a Brownian motion is conditioned to arrive in a fixed submanifold at a fixed positive time. Doing so provides motivation for the remainder of the talk, in which a probabilistic formula for the integral of the heat kernel over a submanifold is proved and used to deduce lower bounds, an asymptotic relation and derivative estimates applicable to the conditioned process.