Abstract
Several characterizations of additive functionals of a Markov process have been described in recent years. Under strong (Hunt) duality hypotheses this was accomplished in a series of papers by Revuz [14], [15], Getoor [9], and Sharpe [17]; for “symmetric” processes this was done by Fukushima [7] and Dynkin [4], [5]; earlier, the situation for Markov stochastic systems was investigated by Dynkin [3], [6]. Here, we obtain results along the same lines for processes in weak duality. The main tool is the “auxiliary process” [13] associated to a pair of Markov processes in weak duality. (Some facts about this process are recalled below.) Our approach is guided in part by similarities with the theory of flows ([8], [16]) and exploits the interplay between optionality and cooptionality in this context.
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References
C. Dellacherie. Ensembles aléatoires I, II. Séminaire de Probabilités III (Univ. Strasbourg), pp. 97–136. Lecture Notes in Math. 88, Springer-Verlag, Berlin, 1967.
C. Dellacherie. Capacités et Processus Stochastiques. Springer-Verlag, New York, 1972.
E.B. Dynkin. Additive functionals of Markov processes and stochastic systems. Ann. Inst. Fourier (Grenoble) 25 (1975), 177–200.
E.B. Dynkin. Additive functionals of several time-reversible Markov processes. J. Functional Analysis 42 (1981), 64–101.
E.B. Dynkin. Green’s and Dirichlet spaces associated with fine Markov processes. (Preprint, 1982)
E.B. Dynkin. Markov systems and their additive functionals. Ann. Probab. 5 (1977), 653–677.
M. Fukushima. Dirichlet Forms and Markov Processes. North-Holland, New York/Kodansha, Tokyo, 1980.
D. Geman and J. Horowitz. Remarks on Palm measures. Arm. Inst. Henri Poinearé, Seo. B. IX (1973), 215–232.
R.K. Getoor. Duality of Lévy systems. Z. Wahrscheinlichkeitstheorie verw. Gebiete 19 (1971), 257–270.
P.A. Meyer. Ensembles aléatoires Markoviens homogènes I, Séminaire de Probabilités III (Univ. Strasbourg), pp. 176–190. Lecture Notes in Math. 381 Springer-Verlag, Berlin, 1974.
J.B. Mitro. Balayage and exit systems for dual Markov processes. (Preprint, 1982)
J.B. Mitro. Dual Markov functionals: applications of a useful auxiliary process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 48 (1979), 97–114.
J.B. Mitro. Dual Markov processes: construction of a useful auxiliary process. Z. Wahrscheinlichkeitstheorie verw. Gebiete 47 (1979), 139–156.
D. Revuz. Measures associées aux fonctionelles additives de Markov I. Trans. Amer. Math. Soc. 148 (1970), 501–531.
D. Revuz. Measures associées aux fonctionelles additives de Markov II. Z. Wahrscheinlichkeitstheorie verw. Gebiete 16 (1970), 336–344.
J. de SamLazaro and P.A. Meyer. Hélices croissantes et mesures de Palm. Séminaire de Probabilités IX (Univ. Strasbourg), pp. 38–51. Lecture notes in Math. 465, Springer-Verlag, Berlin, 1975.
M.J. Sharpe. Discontinuous additive functionals of dual processes. Z. Wahrscheinlichkeitstheorie verw. Gebiete 21 (1975), 81–95.
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© 1983 Birkhäuser, Boston, Inc.
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Atkinson, B.W., Mitro, J.B. (1983). Applications of Revuz and Palm Type Measures for Additive Functionals in Weak Duality. In: Çinlar, E., Chung, K.L., Getoor, R.K. (eds) Seminar on Stochastic Processes, 1982. Progress in Probability and Statistics, vol 5. Birkhäuser Boston. https://doi.org/10.1007/978-1-4684-0540-8_2
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DOI: https://doi.org/10.1007/978-1-4684-0540-8_2
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