Suppose I have a stochastic diffusion $X$. I am studying an expression of the form $u(x):=E^x\int_0^\tau f(X_t)dt$ where $\tau$ is the exit time of $X$ from my bounded open domain $D$. I am also assuming $D$ satisfies the exterior sphere condition.
I want to know what conditions I need to impose on $X$ and $f$ in order to show that $u$ is continuous on $\overline{D}$.
I am already assuming $E^x \tau < \infty$ for all $x$. I am hoping that $f \in L^\infty$ is enough. Intuitively, $u$ should be continuous because a small perturbation to $x + \epsilon \hat v$ could be undone be a random push of approximately $-\epsilon \hat v$ at a very small time. This intuition holds as long as $X$ is diffusive enough, so I expect something like $f\in L^\infty$ and the generator of $X$ is something like $-\triangle + b \cdot \nabla$ where $b$ is bounded to be enough.
If this is a standard theorem somewhere please give references.
Edit: I thought about it some more and I don't think it's true without assuming $f$ is continuous. If $f$ is continuous and bounded the result follows immediately by dominated convergence. If we are working in $\mathbb{R}^n$ for $n>1$ then there doesn't seem to be any way to avoid using a continuity argument.