Capacity theory without duality

Getoor, R. K.; Steffens, Jutta

doi:10.1007/bf00776241

Cited by 8 publications

(7 citation statements)

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“…(b) It is shown in [30] that if v is an entrance law at r, then Qu(a ^ r) = 0, whence the nomenclature.…”

Section: Preliminariesmentioning

confidence: 99%

“…D Let A be as for Corollary (7.15). Then Qm(TA = -oo) < Qm(TA = a) = 0 so that A is ra-cotransient in the terminology of [30]. The measure pa is the cocapacitary measure of A, and its mass pa(1) is the cocapacity of A, denoted C(A).…”

Section: Jdmentioning

confidence: 99%

“…The measure pa is the cocapacitary measure of A, and its mass pa(1) is the cocapacity of A, denoted C(A). See [30]. Let D denote the class of measures p on (E, £) such that p charges no m-polar set, p is carried by A, and pU < ra.…”

Section: Jdmentioning

confidence: 99%

“…The process (Yt, Qm) has proved to be quite useful in studying certain aspects of the process (Xt,Px). See, for example, [1,14,15,16,29,30,36,38,39]. In particular, (Yt,Qm) was used to study the convex cone Exc in [16,19,30].…”

Section: Introductionmentioning

confidence: 99%

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Homogeneous Random Measures and a Weak Order for the Excessive Measures of a Markov Process

Fitzsimmons¹

1987

Transactions of the American Mathematical Society

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ABSTRACT. Let X = (Xt,Px) be a right Markov process and let m be an excessive measure for X. Associated with the pair (X, ra) 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 distribution.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 re such that (Yt,Qç) is obtained by "birthing" (Yt,Qm), birth in [t,t + dt] occurring at rate re(di). Random measures such as re 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. Introduction.Let (Ps : s > 0) be a Borel right semigroup on a Lusin state space (E, £) and let (Xt, Px) denote the associated right continuous strong Markov process. Recall that a measure m on (E, £) is excessive for (Ps) if m is tr-finite and if mPs < m for all s > 0. Since (Ps) is a right semigroup, one then has mPs ] m as s J. 0, as is well known. Let Exc denote the convex cone of excessive measures for (Pa)-Given m E Exc, according to a theorem of Kuznetsov [31], one can construct a stationary process (Yt : t E R) having random birth and death times and a tr-finite governing measure Qm, such that for ii < t2 < • ■ ■ < tn (i» E R)

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“…(b) It is shown in [30] that if v is an entrance law at r, then Qu(a ^ r) = 0, whence the nomenclature.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Jdmentioning

confidence: 99%

Section: Jdmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Homogeneous Random Measures and a Weak Order for the Excessive Measures of a Markov Process

Fitzsimmons¹

1987

Transactions of the American Mathematical Society

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“…In particular, a natural extension of Hunt's balayage LBm was defined in section 5 of [4]. (See also Getoor and Steffens [8,9] and Kaspi [13] for further work on this topic. ) Recall that if the potential kernel U -Psds is proper then any excessive measure m can be realized as the increasing limit of a sequence of potentials.…”

mentioning

confidence: 99%

Penetration times and skorohod stopping

Fitzsimmons

1988

Lecture Notes in Mathematics

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tous droits réservés. L'accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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On a Connection between Kuznetsov Processes and Quasi-Processes

Fitzsimmons

1988

Seminar on Stochastic Processes, 1987

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Capacity theory without duality

Cited by 8 publications

References 16 publications

Homogeneous Random Measures and a Weak Order for the Excessive Measures of a Markov Process

Homogeneous Random Measures and a Weak Order for the Excessive Measures of a Markov Process

Penetration times and skorohod stopping

On a Connection between Kuznetsov Processes and Quasi-Processes

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