Markov operators generated by symmetric measures

With view to applications, we here give an explicit correspondence between the following two: (i) the set of symmetric and positive measures $\rho$ on one hand, and (ii) a certain family of generalized Markov transition measures $P$, with their associated Markov random walk models, on the other. By a generalized Markov transition measure we mean a measurable and measure-valued function $P$ on $(V, \mathcal B)$, such that for every $x \in V , P(x; \cdot)$ is a probability measure on $(V, \mathcal B$). Hence, with the use of our correspondence (i) - (ii), we study generalized Markov transitions $P$ and path-space dynamics. Given $P$, we introduce an associated operator, also denoted by $P$ , and we analyze its spectral theoretic properties with reference to a system of precise $L^2$ spaces. Our setting is more general than that of earlier treatments of reversible Markov processes. In a potential theoretic analysis of our processes, we introduce and study an associated energy Hilbert space $\mathcal H_E$, not directly linked to the initial $L^2$-spaces. Its properties are subtle, and our applications include a study of the $P$-harmonic functions. They may be in $\mathcal H_E$, called finite-energy harmonic functions. A second reason for $\mathcal H_E$ is that it plays a key role in our introduction of a generalized Greens function. (The latter stands in relation to our present measure theoretic Laplace operator in a way that parallels more traditional settings of Greens functions from classical potential theory.) A third reason for $\mathcal H_E$ is its use in our analysis of path-space dynamics for generalized Markov transition systems.