Dirichlet forms on totally disconnected spaces and bipartite Markov chains (Q1303915)

In the general theory of this paper, the state space \(E\) is assumed to be a totally disconnected Lusin space with base \({\mathcal R}\) of the topology of countable simultaneously closed and open sets. Let \(\mu\) and \(\nu\) be probability measures on \(E\) such that \(\text{Supp}[\mu] = E\) and \(k\) a nonnegative Borel function on \(E \times E\). Put \(\Lambda(dx,dy) =k(x,y)\mu(dx)\nu(dy)\) and assume the following conditions: \(\Lambda([(E-R)\times R]\cup [R\times (E-R)])<\infty\) for all \(R \in {\mathcal R}\), \(\int k(x,y)\mu(dx) = \infty\) for a.e. \(y\) with respect to the singular part of \(\nu\) relative to \(\mu\) and there exists \(R_n \in {\mathcal R}\) such that \(\bigcap_{m=n}^\infty R_m\) is compact for all \(n\), \(\sum_{n=1}^\infty \mu(E-R_n)<\infty\) and \(\sum_{n=1}^\infty \Lambda([(E-R_n)\times R_n] \cup [R_n\times (E-R_n)]) <\infty\). The main theorem then says, by proving the quasi-regularity and the tightness of the capacity of the Dirichlet form determined by the closure of \({\mathcal E}(f,g)= \int \int (f(y)-f(x))(g(y)-g(x))\Lambda(dx,dy)\), that there corresponds a \(\mu\)-symmetric Hunt process \((X_t,P_x)\). In particular, if there exists a countable dense subset \(E^0\) of isolated points of \(E\) such that \(\mu\) is concentrated on \(E^0\) and \(\nu\) is concentrated on \(E^*=E-E^0\), then \(X_t\) is called a bipartite Markov chain. The sample path properties such as instantaneous return from the boundary \(E^*\) to the interior \(E^0\) are given. As an example, the rooted tree valued Markov chain with measure \(\mu\) of the distribution of the critical Poisson Galton-Watson branching process is considered. The additive functional and the property of the time changed process on the boundary are also given.