Strongly supermedian kernels and Revuz measures. (English) Zbl 1025.60031

Let \(X\) be a Lusin space, \({\mathcal U}=(U_\alpha)_{\alpha>0}\) a proper sub-Markov resolvent on \(X\) and \(\xi\) an \({\mathcal U}\)-excessive measure. For a strongly supermedian kernel \(V\) on \(X\), define the Revuz measure \(\nu^\xi_V\) by \(\nu^\xi_V(f)=\sup\{\nu(Vf)\); \(\nu\circ U\leq \xi\}\). Further, \(V\) is said regular if, \(Vf \leq s\) on \(\{f>0\}\) for a strongly supermedian function \(s\), then \(Vf \leq s\) everywhere. Analytic characterizations of regular strongly supermedian kernels including Motoo-Mokobodzki property are given. For the excessive measure \(\xi=\rho\circ U+h\) with a harmonic measure \(h\), it is proved that any \(\sigma\)-finite measure charging no set that is both \(\xi\)-polar and \(\rho\)-negligible is a Revuz measure of a regular strongly supermedian kernel. An excessive kernel \(V\) is said natural if, for any Ray open set \(G\) and a function \(f\) vanishing outside of \(G\), \(B^GVf=Vf\), where \(B^GVf\) is the excessive regularization of the réduite of \(Vf\) on \(G\). A characterization of natural excessive kernels by means of the non-charging property of the associated Revuz measure is given. Furthermore, a characterization of the hypothesis (B) of Hunt is given in terms of Revuz measures.


60J40 Right processes
31D05 Axiomatic potential theory
60J45 Probabilistic potential theory
31C15 Potentials and capacities on other spaces
Full Text: DOI


[1] Azéma, J. (1969). Noyau potentiel associé a une fonction excessive d’un processus de Markov. Ann. Inst. Fourier (Grenoble) 19 495-526. · Zbl 0177.21902
[2] Azéma, J. (1973). Théorie générale des processus et retournement du temps. Ann. Sci. École Norm. Sup. 6 459-519. · Zbl 0303.60061
[3] Beznea, L. and Boboc, N. (1994). Duality and biduality for excessive measures. Rev. Roumaine Math. Pures Appl. 39 419-438. · Zbl 0849.31017
[4] Beznea, L. and Boboc, N. (1994). Excessive functions and excessive measures: Hunt’s theorem on balayages, quasi-continuity. NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 430 77-92. · Zbl 0864.31009
[5] Beznea, L. and Boboc, N. (1995). On the integral representation for excessive measures. Rev. Roumaine Math. Pures Appl. 40 725-734. · Zbl 0860.31006
[6] Beznea, L. and Boboc, N. (1996). Kuran’s regularity criterion and localization in excessive structures. Bull. London Math. Soc. 28 273-282. · Zbl 0860.31007
[7] Beznea, L. and Boboc, N. (1996). Quasi-boundedness and subtractivity: applications to excessive measures. Potential Anal. 5 467-485. · Zbl 0869.31013
[8] Beznea, L. and Boboc, N. (1996). Once more about the semipolar sets and regular excessive functions. In Potential Theory-ICPT 94 255-274. de Gruyter, Berlin. · Zbl 0864.31010
[9] Beznea, L. and Boboc, N. (1998). Noyaux fortement surmédians et mesures de Revuz. C. R. Acad. Sci. Paris Sér. I Math. 327 139-142. · Zbl 0917.60069
[10] Beznea, L. and Boboc, N. (1999). Feyel’s techniques on the supermedian functionals and strongly supermedian functions. Potential Anal. 10 347-372. · Zbl 0941.31005
[11] Beznea, L. and Boboc, N. (2000). Excessive kernels and Revuz measures. Probab. Theory Related Fields 117 267-288. · Zbl 0959.60066
[12] Blumenthal, R. M. and Getoor, R. K. (1968). Markov Processes and Potential Theory. Academic, New York. · Zbl 0169.49204
[13] Boboc, N. and Bucur, Gh. (1976). Conuri convexe de funct \?ii continue pe spat\?ii compacte. Editura Academiei, Bucharest. · Zbl 0334.31008
[14] Boboc, N. and Bucur, Gh. (1986). Sur une hypoth ese de G. A. Hunt. C. R. Acad. Sci. Paris Sér. I Math. 302 701-703. · Zbl 0604.60074
[15] Boboc, N., Bucur, Gh. and Cornea, A. (1981). Order and Convexity in Potential Theory: H-Cones. Lecture Notes in Math. 853. Springer, Berlin. · Zbl 0534.31001
[16] Chung, K. L. (1981). Lectures from Markov Processes to Brownian Motion. Springer, Berlin. · Zbl 0503.60073
[17] Dellacherie, C., Maisonneuve, B. and Meyer, P. A. (1992). Probabilités et Potentiel, Chaps. 17-24. Hermann, Paris.
[18] Dellacherie, C. and Meyer, P. A. (1987). Probabilités et Potentiel Chaps. 12-16. Hermann, Paris.
[19] Feyel, D. (1981). Représentation des fonctionnelles surmédianes.Wahrsch. Verw. Gebiete 58 183-198. · Zbl 0475.60059
[20] Fitzsimmons, P. J. (1987). Homogeneous random measures and a weak order for the excessive measures of a Markov process. Trans. Amer. Math. Soc. 303 431-478. · Zbl 0651.60077
[21] Fitzsimmons, P. J. and Getoor, R. K. (1996). Smooth measures and continuous additive functionals of right Markov processes. In It o’s Stochastic Calculus and Probability Theory 31-49. Springer, Berlin. · Zbl 0866.60063
[22] Garcia Alvarez, M. A. (1973). Représentation des noyaux excessifs. Ann. Inst. H. Poincaré Probab. Statist. 9 277-283. · Zbl 0296.60050
[23] Getoor, R. K. (1975). Markov Processes: Ray Processes and Right Processes. Lecture Notes in Math. 440. Springer, Berlin. · Zbl 0299.60051
[24] Getoor, R. K. and Sharpe, M. J. (1984). Naturality, standardness and weak duality for Markov processes.Wahrsch. Verw. Gebiete 67 1-62. · Zbl 0553.60070
[25] Hunt, G. A. (1957). Markoff processes and potentials I. Illinois J. Math. 1 44-93. · Zbl 0100.13804
[26] Meyer, P. A. (1962). Fonctionnelles multiplicatives et additives de Markov. Ann. Inst. Fourier (Grenoble) 12 125-230. · Zbl 0138.40802
[27] Meyer, P. A. (1973). Note sur l’interprétation des mesures d’équilibre. Lecture Notes in Math. 381 191-211. Springer, Berlin.
[28] Mokobodzki, G. (1970). Densité relative de deux potentiels comparables. Lecture Notes in Math. 124 170-194. Springer, Berlin. · Zbl 0218.31014
[29] Mokobodzki, G. (1979). Ensembles compacts de fonctions fortement surmédianes. Lecture Notes in Math. 713 178-193. Springer, Berlin. · Zbl 0416.31013
[30] Mokobodzki, G. (1984). Compactification relative a la topologie fine en théorie du potentiel. Lecture Notes in Math. 1096 450-473. Springer, Berlin. · Zbl 0555.31007
[31] Revuz, D. (1970). Mesures associées aux fonctionelles additives de Markov I. Trans. Amer. Math. Soc. 148 501-531. · Zbl 0266.60053
[32] Sharpe, M. J. (1988). General Theory of Markov Processes. Academic Press, New York. · Zbl 0649.60079
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