Markov Processes Associated With Positivity Preserving Coercive Forms

Ma, Zhi-Ming ; Röckner, Michael

Canadian journal of mathematics, Tome 47 (1995) no. 4, pp. 817-840

Résumé

Coercive closed forms on L 2 -spaces are studied whose associated L 2 -semigroups are positivity preserving. Earlier work by other authors is extended by further developing the potential theory of such forms and completed by proving an analytic characterization of those of these forms which have a probabilistic counterpart, i.e., are associated with (special standard) Markov processes. Examples with finite and infinite dimensional state spaces are discussed.

Détail

DOI : 10.4153/CJM-1995-042-6

Mots-clés : 31C25, 60J40, 60J45, 31C15, positivity preserving coercive forms, Dirichlet forms, quasi-regularity, capacities, Markov processes

Bibliographie

[AMR 92] Albeverio, S., Ma, Z.M. and Rôckner, M., A Beurling-Deny type structure theorem for Dirichlet forms on general state space. Memorial Volume for Hoegh-Krohn, R., Vol. I, Ideas and Methods in Math. Anal., Stochastics and Appl., (eds. Albeverio, S., Fenstad, J.E., Holden, H. and Lindstrom, T.), Cambridge Univ. Press, Cambridge, 1992. Google Scholar

[AMR 93a] Albeverio, S., Quasi-regular Dirichlet forms and Markov processes, J. Funct. Anal. 111(1993), 118–154. Google Scholar

[AMR 93b] Albeverio, S., Local property of Dirichlet forms and diffusions on general state spaces, Math. Ann. 296(1993), 677–686. Google Scholar

[AR 90] Albeverio, S. and Rôckner, M., Classical Dirichlet forms on topological vector spaces — closability and a Cameron-Martin formula, J. Funct. Anal. 88(1990), 395–436. Google Scholar

[AR 91] Albeverio, S., Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms, Probab. Theory Related Fields 89(1991), 347–386. Google Scholar

[An 72a] Ancona, A., Continuité des contractions dans les espaces de Dirichlet, C. R. Acad. Sci. Paris Sér. I 282(1972), 871–873. Google Scholar

[An 72b] Ancona, A., Théorie du potentiel dans les espaces functionnels à forme coercive, Lecture Notes Univ. Paris VI, 1972. Google Scholar

[A 75] Ancona, P.A., Sur les espaces de Dirichlet: principes, fonctions de Green, J. Math. Pures Appl. 54(1975), 75–124. Google Scholar

[BeDe 58] Beurling, A. and Deny, J., Espaces de Dirichlet, Acta Math. 99(1958), 203–224. Google Scholar

[Bl 71] Bliedtner, J., Dirichletforms on regular functional spaces. In: Seminar on Potential Theory II, Lecture Notes in Math. 226, 1–61. Springer, Berlin, 1971. Google Scholar

[BIHa 86] Bliedtner, J. and Hansen, W., Potential Theory, Springer, Berlin, 1986. Google Scholar

[BH 91] Bouleau, J. and Hirsch, F., Dirichlet forms and analysis on Wiener space, de Gruyter, Berlin, New York, 1991. Google Scholar

[Ca-Me 75] Carrillo, S. Menendez, Processus de Markov associé à une forme de Dirichlet non symétrique, Wahrsch. verw. Geb. 33(1975), 139–154. Google Scholar

[Da 89] Davies, E.B., Heat kernels and spectral theory, Cambridge Univ. Press, Cambridge, 1989. Google Scholar

[DM 88] Dellacherie, C. and Meyer, P.A., Probability and Potential Theory, Vol C, Elsevier, Amsterdam, 1988. Google Scholar

[D 84] Doob, J.L., Classical Potential Theory and its Probabilitic Counterpart. Springer, New York, 1984. Google Scholar

[EK 93] Ethier, N.S. and Kurtz, T.G., Fleming-Viotprocesses in population genetics, (1993), preprint. Google Scholar

[F71] Fukushima, M., Dirichlet spaces and strong Markov processes, Trans. Amer. Math. Soc. 162(1971), 185–224. Google Scholar

[F 80] Fukushima, M., Dirichlet forms and Markov processes, North-Holland, Amsterdam, Oxford, New York, 1980. Google Scholar

[Le 77] LeJan, Y., Balayage etformes de Dirichlet, Z. Warsch. verw. Geb. 37(1977), 297–319. Google Scholar

[MOR 93] Ma, Z.M., Overbeck, L. and Rôckner, M., Markov Processes associated with Semi-Dirichlet Forms, Osaka J. Math., to appear. Google Scholar

[MR 92] Ma, Z.M. and Rôckner, M., An Introduction to the theory of (non-symmetric) Dirichlet forms, Berlin, Springer, 1992. Google Scholar

[ORS 93] Overbeck, L., Rôckner, M. and Schmuland, B., An analytic approach to Fleming-Viot processes with interactive selection, Ann. Prob. (1993), to appear. Google Scholar

[S 93] Schmuland, B., On the local property of coercive forms, (1993), preprint. Google Scholar

[Pr 74] Preston, C., Maximum principles and contractions in the potential theory of afunctional Hilbert space, unpublished, (1974), preprint. Google Scholar

[ReS 78] Reed, M. and Simon, B., Methods of modern mathematical physics IV Analysis of Operators, Academic Press, New York, San Francisco, London, 1978. Google Scholar

[RS 93] Rôckner, M. and Schmuland, B., Quasi-regular Dirichlet forms: examples and counterexamples, Canad J. Math. 47(1995), 165–200. Google Scholar

[Sh 88] Sharpe, M.J., General theory of Markov processes, Academic Press, New York, 1988. Google Scholar

[Si 74] Silverstein, M.L., Symmetric Markov Processes, Lecture Notes in Math. 426, Springer, Berlin, Heidelberg, New York, 1974. Google Scholar

[VSC 92] Varopoulos, N.T., Saloff-Coste, L. and Coulhon, T., Analysis and geometry on groups, Cambridge Univ. Press, 1992. Google Scholar