Green's and Dirichlet spaces associated with fine Markov processes

This is the second paper in a series devoted to Green's and Dirichlet spaces. In the first paper, we have investigated Green's space $\text{K}$ and the Dirichlet space $\text{H}$ associated with a symmetric Markov transition function p_t(x, B). Now we assume that p is a transition function of a fine Markov process X and we prove that: (a) the space $\text{H}$ can be built from functions which are right continuous along almost all paths; (b) the positive cone $\text{K}$⁺ in $\text{K}$ can be identified with a cone M of measures on the state space; (c) the positive cone $\text{H}$⁺ in $\text{H}$ can be interpreted as the cone of Green's potentials of measures μ?M. To every measurable set B in the state space E there correspond a subspace $\text{K}$(B) of $\text{K}$ and a subspace $\text{H}$(B) of $\text{H}$. The orthogonal projections of $\text{K}$ onto $\text{K}$ and of $\text{H}$ onto $\text{H}$(B) can be expressed in terms of the hitting probabilities of B by the Markov process X. As the main tool, we use additive functionals of X corresponding to measures μ?M.