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?



Consistency of part processes
What you need is the following fact: the reflecting BM on $\overline{D}$ agrees with the usual BM on $\mathbb{R}^n$ up to the hitting time of $\partial D$. "Agrees" means here that the transition kernels (and resolvents) are the same.


The proof of this fact depends on your favourite definition of the reflecting BM. If you like the energy form (or Dirichlet form) approach, simply note that the energy forms of these processes agree.


