Let Xt be a standard Markov
process on a locally compact separable metric space E having a Radon reference
measure. Let 𝒮 denote the set of locally integrable excessive functions of Xt and ex𝒮
the set of elements lying on the extremal rays of 𝒮. Then if u ∈ ex𝒮 is not
harmonic, it is shown that there is an x ∈ E such that PVu = u for all
neighborhoods V of x where PV is the hitting operator of V . A regularity
condition is introduced which guarantees that two functions in 𝒮 having
the above property at x are proportional. A subset Ê⊂ E and a metric
topology on Ê are defined which allows one to represent each potential
p ∈𝒮 in the form p(x) =∫u(x,y)v(dy) for some finite Borel measure v ≧ 0
on Ê. Here the function u : E ×Ê→ [0,∞] is measurable with respect
to the product Borel field and has the property that for each y ∈Ê the
function x → u(x,y) is an extremal excessive function. In the course of this
study a dual potential operator is introduced and some of its properties are
investigated.
