Abstract
In the context of a transient Borel right Markov process with a fixed excessive measure ξ, we characterize the regular strongly supermedian kernels, producing smooth measures by the Revuz correspondence. In the case of the measures charging no ξ-semipolar sets, this is the analytical counterpart of a probabilistic result of Revuz, Fukushima, and Getoor and Fitzsimmons, concerning the positive continuous additive functionals. We also consider the case of the measures charging no set that is both ξ-polar and ρ-negligible (ρ○U being the potential part of ξ), answering to a problem of Revuz.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Azéma, J.: 'Théorie générale des processus et retournement du temps', Ann. Sci. École Norm. Sup. 6(1973), 459-519.
Beznea, L. and Boboc, N.: 'Excessive functions and excessive measures: Hunt's theorem on balayages, quasi-continuity', in Classical and Modern Potential Theory and Applications, NATO ASI Series C 430, Kluwer Acad. Publ., 1994, pp. 77-92.
Beznea, L. and Boboc, N.: 'Quasi-boundedness and subtractivity: Applications to excessive measures', Potential Anal. 5(1996), 467-485.
Beznea, L. and Boboc, N.: 'Once more about the semipolar sets and regular excessive functions', in Potential Theory-ICPT 94, Walter de Gruyter, 1996, pp. 255-274.
Beznea, L. and Boboc, N.: 'Noyaux fortement surmédianes et mesures de Revuz', C.R. Acad. Sci. Paris Sér. I 327(1998), 139-142.
Beznea, L. and Boboc, N.: 'Feyel's techniques on the supermedian functionals and strongly supermedian functions', Potential Anal. 10(1999), 347-372.
Blumenthal, R.M. and Getoor, R.K.: 'Additive functionals of Markov processes in duality', Trans. Amer. Math. Soc. 112(1964), 131-163.
Dellacherie, C., Maisonneuve, B., and Meyer, P.A.: Probabilités et Potentiel, Ch. XVII-XXIV, Hermann, Paris, 1992.
Dellacherie, C. and Meyer, P.A.: Probabilités et Potentiel, Ch. XII-XVI, Hermann, Paris, 1987.
Fitzsimmons, P.J.: 'Homogeneous random measures and a weak order for the excessive measures of a Markov process', Trans. Amer. Math. Soc. 303(1987), 431-478.
Fitzsimmons, P.J. and Getoor, R.K.: 'Smooth measures and continuous additive functionals of right Markov processes', in Itô's Stochastic Calculus and Probability Theory, Springer, 1996, pp. 31-49.
Fukushima, M.: 'On additive functionals admitting exceptional sets', J. Math. Kyoto Univ. 19(1979), 191-202.
Fukushima, M., Oshima, Y., and Takeda, M.: Dirichlet Forms and Markov Processes, Walter De Gruyter, 1994.
Hirsch, F.: 'Conditions nécessaires et suffisantes d'existence de résolvantes', Z. Warsch. Verw. Gebiete 29(1974), 73-85.
McKean, H.P. and Tanaka, H.: 'Additive functionals of the Brownian path', Mem. Coll. Sci. Univ. Kyoto, Ser. A 33(1961), 479-506.
Meyer, P.A.: 'Fonctionnelles multiplicatives et additives de Markov', Ann. Inst. Fourier (Grenoble) 12(1962), 125-230.
Revuz, D.: 'Mesures associées aux fonctionelles additives de Markov I', Trans. Amer. Math. Soc. 148(1970), 501-531.
Sharpe, M.J.: General Theory of Markov Processes, Academic Press, 1988.
Taylor, J.C.: 'On the existence of sub-Markovian resolvents', Invent. Math. 17(1972), 85-93.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Beznea, L., Boboc, N. Smooth Measures and Regular Strongly Supermedian Kernels Generating Sub-Markovian Resolvents. Potential Analysis 15, 77–87 (2001). https://doi.org/10.1023/A:1011276203807
Issue Date:
DOI: https://doi.org/10.1023/A:1011276203807