I have a question about part processes which appear in theory of Markov processes.

Let $\mathbb{M}=(X_t,P_x)$ be a Markov prccess on a topological space $E$. Let $A $ be an open subset of $E$. The part process $(X_{t}^{A},P_{x})$ of $\mathbb{M}$ on $A$ is defined as follows: \begin{align*} X_{t}^{A}(\omega)&=X_{t}(\omega),\quad t<\tau_{A}(\omega),\\ X_{t}^{A}(\omega)&=\Delta,\quad t \ge \tau_{A}(\omega), \end{align*} where $\Delta$ is a cemetery point and $\tau_{A}(\omega)=\inf\{t>0:X_{t}(\omega) \in E \setminus A\}$.

**Question**: Let us consider the following problem.

Let $D$ be a smooth domain of $\mathbb{R}^d$ and its closure in $\mathbb{R}^d$ is denoted by $\bar{D}$. Let $D_{2} \subset D_{1} \subset \bar{D}$ **be open subsets of $\bar{D}$.** We consider the case that $\partial D \cap D_1 \neq \emptyset$ and $\partial D \cap D_2 \neq \emptyset$. We assume $D_1$ and $D_2$ are also smooth domains.

Let $\mathbb{M}=(X_t,P_x)$ be a reflecting Brownian motion on $\bar{D}$. Then, we can define the part process $\mathbb{M}_{2}=(X_t^{2},P_x)$ of $\mathbb{M}$ on $D_2$.

Let $\mathbb{N}_1=(Y_t,Q_x)$ be a reflecting Brownian motion on $\bar{D}_{1}$. Then, we can define the part process $\mathbb{N}_{2}=(Y_t^{2},Q_x)$ of $\mathbb{N}_1$ on $D_2$.

Do processes $\mathbb{M}_{2}$ and $\mathbb{N}_{2}$ coincide in some sense? For example, semigroups generated by these processes coincide in $dx$-a.e. sense?