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