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


Author: P. J. Fitzsimmons
Journal: Trans. Amer. Math. Soc. 303 (1987), 431-478
MSC: Primary 60J45; Secondary 60G57, 60J55
DOI: https://doi.org/10.1090/S0002-9947-1987-0902778-5
MathSciNet review: 902778
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1987-0902778-5
Article copyright: © Copyright 1987 American Mathematical Society

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