I have a question about Dynkin **Hunt** formula.

Last day, I found a formula in this paper enter link description here.
The formula is the equation (2.5) in this paper, which is called Dynkin **Hunt** formula. I know Dynkin formula. Dynkin formula states
\begin{align*}
R_{\alpha}f(x)=R_{\alpha}^{U}f(x)+E_{x}[e^{-\alpha \tau_{U}}R_{\alpha}f(X_{\tau_{U}})],
\end{align*}
where $\{X_t,P_x\}$ is a Markov(Hunt) process defined on a topological space $E$ and $R_{\alpha}$ is its resolvent. $R_{\alpha}^{U}f(x)$ is defined by
\begin{equation*}
R_{\alpha}^{U}f(x)=E_{x}[\int_{0}^{\tau_{U}}e^{-\alpha t}f(X_t)\,dt],
\end{equation*}
where $\tau_{U}$ is the first exit time from an open set $U$. In other words, $R_{\alpha}^{U}$ is the resolvent of subprocess $(X_{t}^{U},P_x)$ of $(X_t,P_x)$ on $U$.

You can see this formula in *Dirichlet Forms and Symmetric Markov Processes* by M. Fukushima, Y. Oshima, and M. Takeda. The equation (4.1.6) in this book is Dynkin formula.

**Question**

Assume $\{X_t\}$ has transition probability density function $p(t,x,y)$ and transition density function of $\{X_{t}^{U}\}$ is denoted by $p^{U}(t,x,y)$.

Can we show the Dynkin-**Hunt** formula
\begin{equation*}
p(t,x,y)=p^{U}(t,x,y)+E_{x}[1_{\{\tau_{U} \le t\}}p(t-\tau_{U},x,y)]
\end{equation*}
for every $t>0$, $x,y \in E$?

**My attempt**

Assume resolvent density $R_{\alpha}(x,y)$ is continuous in $y$. Then, from Dynkin formula, we see \begin{equation*} R_{\alpha}(x,y)=R_{\alpha}^{U}(x,y)+E_{x}[e^{-\alpha \tau_{U}}r_{\alpha}(X_{\tau_{Y}},y)] \end{equation*} for every $\alpha>0,x,y \in E$. Therefore, \begin{align*} \int_{0}^{\infty}e^{-\alpha t}p(t,x,y)\,dt &=\int_{0}^{\infty}e^{-\alpha t}p^{U}(t,x,y)\,dt \\ &+\int_{0}^{\infty}e^{-\alpha t}E_{x}[1_{\{\tau_{U} \le t\}}p(t-\tau_{U},X_{\tau_{U}},y)]\,dt \end{align*} Therefore, for every $x,y \in E$, \begin{equation*} p(t,x,y)=p^{U}(t,x,y)+E_{x}[1_{\{\tau_{U} \le t\}}p(t-\tau_{U},x,y)] \end{equation*} holds for a.e. $t$. Can we refine this equation? That is, can we show \begin{equation*} p(t,x,y)=p^{U}(t,x,y)+E_{x}[1_{\{\tau_{U} \le t\}}p(t-\tau_{U},x,y)] \end{equation*} for every $t>0$, $x,y \in E$?

**If you know related works about Dynkin-Hunt formula, please let me know.**