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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?

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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.

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  • $\begingroup$ 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? $\endgroup$
    – sharpe
    Commented Sep 6, 2017 at 10:25
  • $\begingroup$ Yes, the Dirichlet form of both processes is $\int_{D_2} |\nabla f|^2$, with domain $W^{1,2}_0(D_2)$. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Sep 6, 2017 at 11:11
  • $\begingroup$ Perhaps you are misunderstanding something... $D_1$ and $D_2$ are open subsets of $\bar{D}$. $\endgroup$
    – sharpe
    Commented Sep 6, 2017 at 11:15
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    $\begingroup$ 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 $\mathbb{R}^n$). $\endgroup$
    – Mateusz Kwaśnicki
    Commented Sep 6, 2017 at 11:34

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