# Consistency of part processes

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?

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.
• There are two Dirichlet forms $\mathcal{E}$, $\mathcal{E}'$ associated with $\mathbb{M}_2$ and $\mathbb{N}_2$. Do you think these Dirichlet forms are the same? Sep 6, 2017 at 10:25
• Yes, the Dirichlet form of both processes is $\int_{D_2} |\nabla f|^2$, with domain $W^{1,2}_0(D_2)$. Sep 6, 2017 at 11:11
• Perhaps you are misunderstanding something... $D_1$ and $D_2$ are open subsets of $\bar{D}$. Sep 6, 2017 at 11:15
• Oh, I see, you mean: open in the relative topology of $\overline{D}$. In this case the answer appears to be still "yes": the domains of the corresponding Dirichlet forms consist of those $f \in W^{1,2}(D_2)$ which vanish on $\partial D_2 \cap D$ (with the boundary taken in the topology of $\mathbb{R}^n$). Sep 6, 2017 at 11:34