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

Abstract

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