Let $X$ be Brownian motion on a Riemannian manifold $M$ starting at $x\in M$, D a domain and $f$ a bounded continuous function on $D$. Define $\tau_D$ to be the first exit time of $X$ from $D$. $u_f\left(t,x,y\right)=\int_Dp_D\left(t,x,y\right)f\left(y\right)dy$ solves
$ \begin{cases} L_Mu_f\left(t,x\right)=0, & t>0,x\in\overline{D}, \\ u_f\left(t,x\right)=0, & t>0,x\in\partial D, \\ \lim_{t\downarrow0}u_f\left(t,x\right)=f\left(x\right), & x\in D. \end{cases} $
I have seen the following two formulae written:
$$\mathbb{P}_x\left(X_t\in C, t<\tau_D\right)=\int_Cp_D\left(t,x,y\right)dy$$ and $$E_x\left(f\left(X_t\right),t<\tau_D\right)=\int_Dp_D\left(t,x,y\right)f\left(y\right)dy.$$
Is $\mathbb{P}_x$ the joint probability of the events $\{X_t\in C\}$ and $\{t<\tau_D\}$? I also do not know what the interpretation for $E_x\left(f\left(X_t\right),t<\tau_D\right)$ is. Any help is appreciated.
Nevermind, it turns out that $\mathbb{P}_x$ is as I thought and $\mathbb{E}_x\left(f\left(X_t\right),t<\tau_D\right)=\mathbb{E}_x\left(f\left(X_t\right)I_{\{t<\tau_D\}}\right)$.