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Quasi-Regular Dirichlet Forms: Examples and Counterexamples

Published online by Cambridge University Press:  20 November 2018

Michael Röckner  and
Byron Schmuland
Michael Röckner
Affiliation:
Institut für Angewandte Mathematik Universität Bonn Wegelerstraβe 6 53115 Bonn Germany
Byron Schmuland
Affiliation:
Department of Statistics and Applied Probability University of Alberta Edmonton, Alberta T6G 2G1
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Abstract

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We prove some new results on quasi-regular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of examples including cases with possibly degenerate (sub)-elliptic part, diffusions on loop spaces, and certain Fleming- Viot processes.

Keywords

31C2560J60Dirichlet formsquasi-regularitycapacitiessmooth measuresexceptional setstightnesssquare field operatorM processesdegenerate cases
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 47 , Issue 1 , 01 February 1995 , pp. 165 - 200
Copyright
Copyright © Canadian Mathematical Society 1995

References

[AH-K 75] Albeverio, S. and Høegh-Krohn, R., Quasi-invariant measures, symmetric diffusion processes and quantum fields. In: Les méthodes mathématiques de la théorie quantique des champs, Colloques Internationaux du C.N.R.S. 248, Marseille, 23–27.juin 1975, C.N.R.S., 1976.Google Scholar
[AH-K 77al Albeverio, S., Dirichletforms and diffusion processes on rigged Hubert spaces, Z. Wahrsch. verw. Geb. 40(1977), 1–57.Google Scholar
[AH-K 77b] Albeverio, S., Hunt processes and analytic potential theory on rigged Hubert spaces, Ann. Inst. H.Poincaré Sect. (B) 13(1977), 269–291.Google Scholar
[ALR 931 Albeverio, S., Léandre, R. and Röckner, M., Construction of a rotational invariant diffusion on the free loop space, C.R. Acad. Sci. Paris 316(1993), 287–292.Google Scholar
[AM 91a] Albeverio, S. and Ma, Z.M., Diffusion processes associated with singular Dirichlet forms, In: Proc. Stochastic Anal. Appl., (eds. Cruzeiro, A.B. and Zambrini, J.C.), Birkhauser, New York, 1991. 11–28.Google Scholar
[AM 91b] Albeverio, S., Perturbation of Dirichlet forms: Lower semiboundedness, closability, and form cores,, J. Funct. Anal. 99(1991), 332–356.Google Scholar
[AMR 93a] Albeverio, S., Ma, Z.M. and R, M.öckner, 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
[AMR 93c] Albeverio, S., A remark on quasi-Radon measures, Proceedings of the 3rd Locarno Conference 1991, World Scientific, (1993), preprint.Google Scholar
[AR 89] Albeverio, S. and R, M.öckner, Classical Dirichlet forms on topological vector spaces: The construction of the associated diffusion process, Probab. Theory Related Fields 83(1989), 405–434.Google Scholar
[An 76] Ancona, A., Continuité des contractions dans les espaces de Dirichlet. In: Sem. Théor. Potentiel. Paris (2), Lecture Notes in Math. 563,1-26, Springer, Berlin, 1976.Google Scholar
[Bl 71] Bliedtner, J., Dirichlet forms on regular functional spaces, In: Sem. Potential Theory II, Lecture Notes in Mathematics 226, 14-61. Springer, Berlin, 1971.Google Scholar
[Bou 74] Bourbaki, N., Topologie Générale, Chapitres 5 À 10, Hermann, Paris, 1974.Google Scholar
[CaMe 75] Carrillo Menendez, S., Processus de Markov associé a une forme de Dirichlet non symétrique Z. Wahrsch. verw. Geb. 33(1975), 139-154.Google Scholar
[Da 89] Davies, E. B., Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.Google Scholar
[DR 92] Driver, B. and RÔckner, M., Construction of diffusions on path and loop spaces of compact Riemannian manifolds, C. R. Acad. Sci. Paris 315(1992), 603-608.Google Scholar
[EK 93] Ethier, S. N. and Kurtz, T. G., Fleming-Viot processes in population genetics, SI AM J. Control Optim. 31(1993), 345-386.Google Scholar
[F80] Fukushima, M., Dirichlet Forms and Markov Processes, Amsterdam, North Holland, 1980.Google Scholar
[FST 91] Fukushima, M., Sato, K. and Taniguchi, S., On the closable parts of pre-Dirichlet forms and the fine support of underlying measures, Osaka J. Math. 28(1991), 517-535.Google Scholar
[JL 91] Jones, J. D. S. and R. Léandre, Lp-Chenforms on loop spaces. In: Stochastic Analysis, (eds. Barlow, M. and Bingham, N.), Cambridge, Cambridge University Press, 1991, 103-163.Google Scholar
[Ku 821 Kusuoka, S., Dirichlet forms and diffusion processes on Banach spaces, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29(1982), 79-95.Google Scholar
[L 92] Léandre, R., Invariant Sobolev calculus on the free loop space, (1992), preprint.Google Scholar
[L 93] Léandre, R., Integration by parts formulas and rotationally invariant Sobolev calculus on free loop spaces, Infinite-dimensional geometry in physics, Karpacz, 1992, J. Geom. Phys. 11(1993), 517-528.Google Scholar
[LM 72] Lions, J. L. and Magenes, E., Non-Homogeneous Boundary Value Problems and Applications, Volume I, Grundlehren Math. Wiss. 181, Springer, Berlin, 1972.Google Scholar
[LR 92] Lyons, T. and Rôckner, M., A note on tightness of capacities associated with Dirichlet forms, Bull. London Math. Soc. 24(1992), 181-184.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., Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer, Berlin, 1992.Google Scholar
[O88] Oshima, Y., Lectures on Dirichlet forms, Erlangen, (1988), preprint.Google Scholar
[ORS 93] Overbeck, L., Rôckner, M. and Schmuland, B., An analytic approach to Fleming-Viot processes with interactive selection, Ann. Probab., to appear.Google Scholar
[RS 92] Rôckner, M. and Schmuland, B., Tightness of general C\p capacities on Banach space, J. Funct. Anal. 108(1992), 1-12.Google Scholar
[RW 85] Rôckner, M. and Wielens, N., Dirichlet forms: Closability and change of speed measure. In: Infinite dimensional analysis and stochastic processes, (ed. S. Albeverio), Research Notes in Math. 124, Pitman, Boston, London, Melbourne, 1985, 119-144.Google Scholar
[S90] Schmuland, B., An alternative compactification for classical Dirichlet forms on topological vector spaces, Stochastics Stochastics Rep. 33(1990), 75-90.Google Scholar
[St 65] Stampacchia, G., Le problme de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier 15(1965), 189-258.Google Scholar
[Sto 93] Stollmann, P., Closed ideals in Dirichlet spaces, 1993, preprint.Google Scholar
[StoV 93] Stollmann, P. and Voigt, J., Perturbation of Dirichlet forms by measures, 1993, preprint.Google Scholar
[Str 88] Stroock, D. W., Diffusion semigroups corresponding to uniformly elliptic divergence form operators. In: Séminaire de Probabilités XXII, (eds. Azéma, J., Meyer, P. A. and Yor, M.), Lecture Notes in Math. 1321, Springer, Berlin, 1988, 316-347.Google Scholar
[Stu 93] Sturm, K. T., Schrôdinger operators with arbitrary non-negative potentials, Oper. Theory Adv. Appl. 57(1992), 291-306.Google Scholar
[VSC 92] Varopoulos, N. Th., Saloff-Coste, L. and Coulhon, T., Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992.Google Scholar