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

Abstract:

Let $X = ({X_t}, {P^x})$ 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 $({Y_t}, {Q_m})$ with random times of birth and death, with the same transition function as $X$, and with $m$ as one dimensional distribution. We use $({Y_t}, {Q_m})$ to study the cone of excessive measures for $X$. A "weak order" is defined on this cone: an excessive measure $\xi$ is weakly dominated by $m$ if and only if there is a suitable homogeneous random measure $\kappa$ such that $({Y_t}, {Q_\xi })$ is obtained by "birthing" $({Y_t}, {Q_m})$, birth in $[t, t + dt]$ occurring at rate $\kappa (dt)$. Random measures such as $\kappa$ are studied through the use of Palm measures. We also develop aspects of the "general theory of processes" over $({Y_t}, {Q_m})$, including the moderate Markov property of $({Y_t}, {Q_m})$ when the arrow of time is reversed. Applications to balayage and capacity are suggested.