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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Homogeneous random measures and a weak order for the excessive measures of a Markov process
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by P. J. Fitzsimmons PDF
Trans. Amer. Math. Soc. 303 (1987), 431-478 Request permission

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.
References
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Additional Information
  • © Copyright 1987 American Mathematical Society
  • 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