Homogeneous random measures and a weak order for the excessive measures of a Markov process

Let X = (Xt,PX) be a right Markov process and let m be an excessive measure for X. Associated with the pair (X, m) is a stationary strong Markov process (Yt, Qm) with random times of birth and death, with the same transition function as X, and with m as one dimensional di^stribution. We use (Yt,Qm) to study the cone of excessive measures for X. A "weak order" is defined on this cone: an excessive measure ( is weakly dominated by m if and only if there is a suitable homogeneous random measure tt such that (Yt, Q6) is obtained by "birthing" (Yt, Qm), birth in [t, t + dt] occurring at rate s(dt). Random measures such as s are studied through the use of Palm measures. We also develop aspects of the "general theory of processes" over (Yt,Qm), including the moderate Markov property of (Yt, Qm) when the arrow of time is reversed. Applications to balayage and capacity are suggested.

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