Irreducible decomposition for Markov processes (Q1979909)

Let \((\mathscr{E},\mathscr{F})\) be a quasi-regular semi-Dirichlet form with a lower bound on \(L^2(E,m)\) properly associated with a diffusion or an \(m\)-symmetric Markov process \(X\). For \(D\subset E\), \(\sigma_D:=\inf\{t>0:X_t\in D\}\). A nearly Borel set \(A\subset E\) is called finely open if for any \(x\in A\), \(\mathbf{P}_x(\sigma_{E\setminus A}>0)=1\). The complement of a finely open set is called finely closed. Assume that \(X\) satisfies the absolute continuity condition with respect to the transition probability, i.e., \(P_t(x,dy)\ll m(dy)\) for each \(x\in E\) and \(t\geq 0\), where \(P_t\) is the transition kernel of \(X\). The main result of this paper obtains the following irreducible decomposition for \(X\): For each \(x\in E\), there exists a unique finely open and finely closed set \(E_x\) satisfying that \begin{itemize} \item[(1)] \(x\in E_x\). \item[(2)] \(E_x\) and its complement \(E_x^c\) are \(X\)-invariant in the sense that \[ \mathbf{P}_x\left(X_t\in B, \forall t\in [0,\zeta[,\; X_{t-}\in B, \forall t\in ]0, \zeta[\right)=1,\quad x\in B \] where \(B=E_x\) or \(E_x^c\) and \(\zeta\) is the life time of \(X\). \item[(3)] The part process \(X_{E_x}\) of \(X\) is finely irreducible in the sense that for any finely open nearly Borel set \(D\) with \(D\neq \emptyset\), \(\mathbf{P}_x(\sigma_D<\infty)>0\) for all \(x\in E_x\). \end{itemize} More strongly, there exist at most countably many sets \(\{x_i\}\) such that \(E=\bigcup_{i=1}^N E_{x_i}\), where \(N\in \mathbb{N}\) or \(N=\infty\), forms a disjoint union. Particularly, if \(X\) satisfies the strong Feller property, then \(E_x\) can be taken to be open and closed.