Submanifold bridge processes

We introduce and study submanifold bridge processes. Our method involves proving a general formula for the integral over a submanifold of the minimal heat kernel on a complete Riemannian manifold. Our formula expresses this object in terms of a stochastic process whose trajectories terminate on the submanifold at a fixed positive time. We study this process and use the formula to derive lower bounds, an asymptotic relation and derivative estimates. Using these results we introduce and characterize Brownian bridges to submanifolds. Before doing so we prove necessary estimates on the Laplacian of the distance function and define a notion of local time on a hypersurface. These preliminary developments also lead to a study of the distance between Brownian motion and a submanifold, in which we prove exponential bounds and concentration inequalities. This work is motivated by the desire to extend the analysis of path and loop space to measures on paths which terminate on a submanifold