Let $X$ be a countable set. Let $c:X\times X\to[0,+\infty)$ satisfy $$c(x,y)=c(y,x)\text{ for all }x,y\in X,$$ $$m(x)=\sum_{y\in X}c(x,y)\in (0,+\infty)\text{ for all }x\in X,$$ $$c(x,x)=0\text{ for all }x\in X.$$
(1) Let $P(x,y)={c(x,y)}/{m(x)} \text{ for all }x,y\in X$. Then we can have a reversible Markov chain on $X$ with transition probability $P$.
(2) Regarding $m$ as a measure on $X$, we have the space $L^2(X;m)$.
Let $C_0(X)=\{u:\text{real function on }X \text{ with } \{x:u(x)\ne0\} \text{ is finite}\}$. We can define $$ \begin{aligned} \mathscr{E}(u,v)&=\frac{1}{2}\sum_{x,y\in X}c(x,y)(u(x)-u(y))(v(x)-v(y)),\\ \mathscr{D}[\mathscr{E}]&=\text{ the }\mathscr{E}_1\text{-closure of }C_0(X). \end{aligned} $$ Then $(\mathscr{E},\mathscr{D}[\mathscr{E}])$ is a regular Dirichlet form on $L^2(X;m)$ which corresponds to a Hunt process.
So what is the relation between the Markov chain in (1) and the Hunt process in (2)?
