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 Electronic Journal of Probability > Vol. 8 (2003) > Paper 10 open journal systems 


Homogeneous Random Measures and Strongly Supermedian Kernels of a Markov Process

Patrick J. Fitzsimmons, UCSD
Ronald K. Getoor, UCSD


Abstract
The potential kernel of a positive left additive functional (of a strong Markov process X) maps positive functions to strongly supermedian functions and satisfies a variant of the classical domination principle of potential theory. Such a kernel V is called a regular strongly supermedian kernel in recent work of L. Beznea and N. Boboc. We establish the converse: Every regular strongly supermedian kernel V is the potential kernel of a random measure homogeneous on [0, infinity). Under additional finiteness conditions such random measures give rise to left additive functionals. We investigate such random measures, their potential kernels, and their associated characteristic measures. Given a left additive functional A (not necessarily continuous), we give an explicit construction of a simple Markov process Z whose resolvent has initial kernel equal to the potential kernel UA. The theory we develop is the probabilistic counterpart of the work of Beznea and Boboc. Our main tool is the Kuznetsov process associated with X and a given excessive measure m.


Full text: PDF

Pages: 1-54

Published on: July 3, 2003
Bibliography
  1. Azéma, J.: Noyau potentiel associé à une fonction excessive d'un processus de Markov. Ann. Inst. Fourier (Grenoble) 19 (1969) 495-526. [MR: 43#1271]
  2. Azéma, J.: Une remarque sur les temps de retour. Trois applications. In Séminaire de Probabilités, VI. Springer, Berlin, 1972, pp. 35-50. Lecture Notes in Math., Vol. 258. [MR: 52#1907]
  3. Azéma, J.: Théorie générale des processus et retournement du temps. Ann. Sci. École Norm. Sup. (4) 6 (1973) 459-519. [MR: 51#1977 ]
  4. Beznea, L., and Boboc, N.: Feyel's techniques on the supermedian functionals and strongly supermedian functions. Potential Anal. 10 (1999) 347-372. [MR: 2000f:60114]
  5. Beznea, L., and Boboc, N.: Excessive kernels and Revuz measures. Probab. Theory Related Fields 117 (2000) 267-288. [MR: 2001h:60131]
  6. Beznea, L., and Boboc, N.: Smooth measures and regular strongly supermedian kernels generating sub-Markovian resolvents. Potential Anal. 15 (2001) 77-87. [MR: 2003f:60140]
  7. Beznea, L., and Boboc, N.: Strongly supermedian kernels and Revuz measures. Ann. Probab. 29 (2001) 418-436. [MR: 2002e:60120]
  8. Beznea, L., and Boboc, N.: Fine densities for excessive measures and the Revuz correspondence. Preprint (2002).
  9. Blumenthal, R.M. and Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, New York, 1968. [MR: 41#9348 ]
  10. Dellacherie, C.: Capacités et Processus Stochastiques. Springer-Verlag, Berlin, 1972. [MR: 56#6810]
  11. Dellacherie, C.: Autour des ensembles semi-polaires. In Seminar on Stochastic Processes, 1987. Birkhäuser Boston, Boston, MA, 1988, pp. 65-92. [MR: 91i:60191]
  12. Dellacherie, C. and Meyer, P.-A.: Probabilités et Potentiel. Chapitres V à VIII. Hermann, Paris, 1980. Théorie des martingales. [MR: 82b:60001]
  13. Dellacherie, C. and Meyer, P.-A.: Probabilités et Potentiel. Chapitres IX à XI. Hermann, Paris, 1983. Théorie discrète du potential. [MR: 86b:60003]
  14. Dellacherie, C. and Meyer, P.-A.: Probabilités et Potentiel. Chapitres XII-XVI. Hermann, Paris, 1987. Théorie du potentiel associée à une résolvante. Théorie des processus de Markov. [MR: 88k:60002 ]
  15. Dellacherie, C., Meyer, P.-A. and Maisonneuve, B.: Probabilités et Potentiel. Chapitres XVII-XXIV. Hermann, Paris, 1992. Processus de Markov (fin), Compléments de calcul stochastique.
  16. Duncan, R.D. and Getoor, R.K.: Equilibrium potentials and additive functionals. Indiana Univ. Math. J. 21 (1971/1972) 529-545. [MR: 44#6035]
  17. Dynkin, E.B.: Markov Processes, Vols. I,II. Academic Press, New York, 1965. [MR: 33#1887]
  18. Dynkin, E.B.: Additive functionals of Markov processes and stochastic systems. Ann. Inst. Fourier (Grenoble) 25 (1975) 177-200. [MR: 55#9294]
  19. Dynkin, E.B.: Markov systems and their additive functionals. Ann. Probab. 5 (1977) 653-677. [MR: 56#9701]
  20. Feyel, D.: Représentation des fonctionnelles surmédianes. Z. Wahrsch. verw. Gebiete 58 (1981) 183-198. [MR: 82m:31013]
  21. Feyel, D.: Sur la théorie fine du potentiel. Bull. Soc. Math. France 111 (1983) 41-57. [MR: 85d:31017]
  22. 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. [MR: 89c:60088]
  23. Fitzsimmons, P. J.: Markov processes and nonsymmetric Dirichlet forms without regularity. J. Funct. Anal. 85 (1989) 287-306. [MR: 90i:60053]
  24. Fitzsimmons, P.J. and Getoor, R.K.: A fine domination principle for excessive measures. Math. Z. 207 (1991) 137-151. [MR: 93e:60157]
  25. Fitzsimmons, P.J. and Getoor, R.K.: A weak quasi-Lindelöf property and quasi-fine supports of measures. Math. Ann. 301 (1995) 751-762. [MR: 96e:60138]
  26. 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, Tokyo, 1996, pp. 31-49. [MR: 98g:60137]
  27. Fukushima, M., Oshima, Y., and Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter & Co., Berlin, 1994. [MR: 96f:60126]
  28. Getoor, R.K.: Excessive Measures. Birkhäuser, Boston, 1990. [MR: 92i:60135]
  29. Getoor, R.K.: Measures not charging semipolars and equations of Schrödinger type. Potential Anal. 4 (1995) 79-100. [MR: 96b:60197]
  30. Getoor, R.K.: Measure perturbations of Markovian semigroups. Potential Anal. 11 (1999) 101-133. [MR: 2001c:60119]
  31. Getoor, R.K. and Sharpe, M.J.: Balayage and multiplicative functionals. Z. Wahrsch. verw. Gebiete 28 (1973/74) 139-164. [MR: 51#9229]
  32. Getoor, R.K. and Sharpe, M.J.: Naturality, standardness, and weak duality for Markov processes. Z. Wahrsch. verw. Gebiete 67 (1984) 1-62. [MR: 86f:60093]
  33. Hirsch, F.: Conditions nécessaires et suffisantes d'existence de résolvantes. Z. Wahrsch. verw. Gebiete 29 (1974) 73-85. [MR: 51#7011]
  34. Hunt, G.A.: Markoff processes and potentials. I,II. Illinois J. Math. 1 (1957) 44-93, 316-369. [MR: 19,951g]
  35. Mertens, J.-F.: Strongly supermedian functions and optimal stopping. Z. Wahrsch. verw. Gebiete 26 (1973) 119-139. [MR: 49#11617]
  36. Meyer, P.-A.: Fonctionelles multiplicatives et additives de Markov. Ann. Inst. Fourier (Grenoble) 12 (1962) 125-230. [MR: 25#3570]
  37. Mokobodzki, G.: Compactification relative à la topologie fine en théorie du potentiel. In Théorie du potentiel (Orsay, 1983). Springer, Berlin, 1984, pp. 450-473. [MR: 88j:31011]
  38. Revuz, D.: Mesures associées aux fonctionnelles additives de Markov. I. Trans. Amer. Math. Soc. 148 (1970) 501-531. [MR: 43#5611]
  39. Revuz, D.: Mesures associées aux fonctionnelles additives de Markov. II. Z. Wahrsch. verw. Gebiete 16 (1970) 336-344. [MR: 43#6979]
  40. Rost, H.: The stopping distributions of a Markov Process. Invent. Math. 14 (1971) 1-16. [MR: 49#11641]
  41. Sharpe, M.J.: Exact multiplicative functionals in duality. Indiana Univ. Math. J. 21 (1971/72) 27-60. [MR: 55#9287]
  42. Sharpe, M.: General Theory of Markov Processes. Academic Press, Boston, 1988. [MR: 89m:60169]
  43. Sur, M.G.: Continuous additive functionals of Markov processes and excessive functions. Soviet Math. Dokl. 2 (1961) 365-368. [MR: 26#7023 ]
  44. Taylor, J.C.: On the existence of sub-Markovian resolvents. Invent. Math. 17 (1972) 85-93. [MR: 49#9961]
  45. Taylor, J.C.: A characterization of the kernel limlambda --> 0 Vlambda for sub-Markovian resolvents (Vlambda). Ann. Probab. 3 (1975) 355-357. [MR: 51#9227]
  46. Volkonskii, V.A.: Additive functionals of Markov processes. Trudy Moskov. Mat. Obsc. 9 (1960) 143-189. [MR: 25#610 ]
















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Electronic Journal of Probability. ISSN: 1083-6489