consider a complete Riemannian manifold $M$ with heat kernel $p\_M$ and let $U\subset M$ be an open set. Let $W\_{x,t}^{y}$ be the Wiener measure associated to the Brownian motion starting at $x$ and ending at $y$ after time 't'. Consider the following function:


$$U\ni y \mapsto E\_{t}^{x,y}\left( 1\_{\{ t<\tau\_U \}} \right)\in [0,1],$$


where $\tau\_U$ is the first exit time from $U$. I am wondering if it is known whether this function is continuous or not? Is there a reference where I can find this proven in the general case described above?


I would very appreciate any help!


Best wishes



Conditional Wiener measure continuous
As the transition densities for the Wiener measure are continuous, weak continuity results of a type that may be what you seek can be found in <http://projecteuclid.org/euclid.aop/1298669175> ["Markovian bridges: Weak continuity and pathwise constructions" by L. Chaumont and G. Uribe Bravo].


