Vol. 39, No. 1, 1971

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Integral representation of excessive functions of a Markov process

Richard D. Duncan

Vol. 39 (1971), No. 1, 125–144

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 PV u = 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.

Mathematical Subject Classification 2000
Primary: 60J45
Received: 1 September 1970
Published: 1 October 1971
Richard D. Duncan