Duality and integral representation for excessive measures. (English) Zbl 0770.60069
The author generalizes results that treat the integral representation of excessive measures with respect to a given substochastic resolvent. She proves that there exists a unique integral representation over extremal excessive measures if either the underlying state space is a Radon space or the excessive measure is supported on a standard Borel subset or the resolvent is proper and absolutely continuous with respect to a \(\sigma\)- finite measure. For the proof the excessive measure is decomposed into a dissipative and a conservative measure and each case is handled separately, the first one using a duality result that characterizes the linear functionals induced by the energy functional and all bounded excessive measures.
Reviewer: W.Hoh (Erlangen)
MSC:
| 60J45 | Probabilistic potential theory |
| 46A55 | Convex sets in topological linear spaces; Choquet theory |
Keywords:
integral representation of excessive measures; extremal excessive measures; decomposed into a dissipative and a conservative measure; energy functionalReferences:
| [1] | Alfsen, E.M.: Compact Convex Sets and Boundary Integrals. Berlin Heidelberg New York: Springer 1971 · Zbl 0209.42601 |
| [2] | Asimow, L., Ellis, A.I.: Convexity Theory and Its Applications in Functional Analysis. New York: Academic Press 1980 · Zbl 0453.46013 |
| [3] | Ben Saad, H., Janßen, K.: Bernstein’s theorem for completely excessive measures. Nagoya Math. J.119, 133–141 (1990) |
| [4] | Beznea, L.: Ultrapotentials and positive eigenfunctions for an absolutely continuous resolvent of kernels. Nagoya Math. J.112, 125–142 (1988) · Zbl 0632.31008 |
| [5] | Blumenthal, R.M.: A decomposition of excessive measures. In: Çinlar, E. et al. (eds.) Sem. Stochastic Processes 1985, pp. 1–8. Boston: Birkhäuser 1986 · Zbl 0602.60061 |
| [6] | blumenthal, R.M., Getoor, R.K.: Markov Processes and Potential Theory. New York London: Academic Press 1968 · Zbl 0169.49204 |
| [7] | Boboc, N., Bucur, Gh., Cornea, A.: Order and Convexity in Potential Theory:H-Cones. (Lect. Notes Math., vol 853) Berlin Heidelberg New York: Springer 1981 · Zbl 0492.31010 |
| [8] | Cornea, A., Licea, G.: Order and Potential. Resolvent Families of Kernels. (Lect. Notes Math., vol. 494) Berlin Heidelberg New York: Springer 1975 · Zbl 0308.31002 |
| [9] | Dellacherie, C., Meyer, P.A.: Probabilités et potentiel, Chap I–IV. Paris: Hermann 1975 · Zbl 0323.60039 |
| [10] | Dellacherie, C., Meyer, P.A.: Probabilités et potentiel, Chap. IX–XI. Théorie discrète du potentiel. Paris: Hermann 1983 |
| [11] | Dellacherie, C., Meyer, P.A.: Probabilités et potentiel, Chap. XII–XVI. Théorie du potentiel associée á une résolvante. Théorie des processus de Markov. Paris: Hermann 1987 |
| [12] | Dubins, L.E., Freedman, D.A.: Exchangeable processes need not be mixtures of independent, identically distributed random variables. Z. Wahrscheinlichkeitstheor. Verw. Geb.48, 115–132 (1979) · Zbl 0388.60036 · doi:10.1007/BF01886868 |
| [13] | Dynkin, E.B.: Integral representation of excessive measures and excessive functions. Russ. Math. Surv.27(1), 43–84 (1972) · Zbl 0321.60002 · doi:10.1070/RM1972v027n01ABEH001363 |
| [14] | Dynkin, E.B.: Minimal excessive functions and measures. Trans. Am. Math. Soc.258, 217–244 (1980) · Zbl 0422.60057 · doi:10.1090/S0002-9947-1980-0554330-5 |
| [15] | Fitzsimmons, P.J., Maisonneuve, B.: Excessive measures and Markov processes with random birth and death. Z. Wahrscheinlichkeitstheor. Verw. Geb.72, 319–336 (1986) · Zbl 0584.60085 |
| [16] | Getoor, R.K.: Markov Processes: Ray Processes and Right Processes. (Lect. Notes Math., vol. 440) Berlin Heidelberg New York: Springer 1975 · Zbl 0299.60051 |
| [17] | Getoor, R.K.: Excessive Measures. Lecture Notes. Boston: Birkhäuser 1990 · Zbl 0982.31500 |
| [18] | Getoor, R.K., Glover, J.: Riesz decompositions in Markov process theory. Trans. Am. Math. Soc.185, 107–132 (1984) · Zbl 0547.60076 · doi:10.1090/S0002-9947-1984-0748833-1 |
| [19] | Getoor, R.K., Steffens, J.: The energy functional, balayage, and capacity. Ann. Inst. Henri Poincaré23, 321–357 (1987) · Zbl 0617.60073 |
| [20] | Halmos, P.R.: Measure Theory. London: Van Nostrand 1950 · Zbl 0040.16802 |
| [21] | Kerstan, J., Wakolbinger, A.: Ergodic decomposition of probability laws. Z. Wahrscheinlichkeitstheor. Verw. Geb.56, 399–414 (1981) · Zbl 0444.60004 · doi:10.1007/BF00536180 |
| [22] | Kunita, H., Watanabe, T.: Markov processes and Martin boundaries I. III. J. Math.9, 485–526 (1965) · Zbl 0147.16505 |
| [23] | Kunita, H., Watanabe, T.: Some theorems concerning resolvents over locally compact spaces. In: Proc. 5th Berkeley Symp. II.2, pp. 131–164, Berkeley, Calif.: University of California Press 1967 · Zbl 0221.60044 |
| [24] | Martin, R.S.: Minimal positive harmonic functions. Trans. Am. Math. Soc.49, 127–172 (1941) · Zbl 0025.33302 |
| [25] | Meyer, P.A.: Processus de Markov: la frontière de Martin. (Lect. Notes Math., vol. 77) Berlin Heidelberg New York: Berlin 1968 · Zbl 0174.49303 |
| [26] | Meyer, P.A.: Réprésentation intégrale des fonctions excessives. Résultats de Mokobodzki. In: Meyer, P.A. (ed.) Sém. Probab. V. (Lect. Notes Math., vol. 191, pp. 196–208) Berlin Heidelberg New York: Springer 1971 |
| [27] | Meyer, P.A.: Le schéma de remplissage en temps continus d’après H. Rost. In: Meyer, P.A. (ed.) Sém. Probab. VI, pp. 130–150. Berlin Heidelberg New York: Springer 1972 |
| [28] | Mokobodzki, G.: Réprésentation intégrale des fonctions surharmoniques au moyen des réduites. Ann. Inst. FourierXV, 103–112 (1965) · Zbl 0134.09502 |
| [29] | Mokobodzki, G.: Structure des cônes de potentiels. Sémin. Bourbaki377 (1969/70) |
| [30] | Mokobodzki, G.: Dualité formelle et réprésentation intégrale des fonctions excessives. Actes, Congrès Intern. Math. 1970,2, 531–535 (1971) |
| [31] | Phelps, R.R.: Lectures on Choquet’s Theorem. London: Van Nostrand 1966 · Zbl 0135.36203 |
| [32] | Rogers, L.C.G.: Ito excursion theory via resolvents. Z. Wahrscheinlichkeitstheor. Verw. Geb.63, 237–255 (1983) · Zbl 0528.60073 · doi:10.1007/BF00538964 |
| [33] | Sharpe, M.: General Theory of Markov Processes. San Diego: Academic Press 1988 · Zbl 0649.60079 |
| [34] | Steffens, J.: Excessive measures and the existence of right semigroups and processes. Trans. Am. Math. Soc.311, 267–290 (1989) · Zbl 0658.60108 · doi:10.1090/S0002-9947-1989-0929664-0 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
