×

Gauge theorems for resolvents with application to Markov processes. (English) Zbl 0725.60080

Let \({\mathbb{V}}=(V_{\alpha})_{\alpha \geq 0}\) be a (not necessarily sub- Markovian) resolvent such that the kernel \(V_{\alpha}\) for some \(\alpha\geq 0\) is compact and irreducible. We prove the following general gauge theorem: If there exists at least one \({\mathbb{V}}\)-excessive function which is not \({\mathbb{V}}\)-invariant, then \(V_ 0\) is bounded. This result will be applied to resolvents \({\mathbb{U}}^ M\) arising from perturbation of sub-Markovian right resolvents \({\mathbb{U}}\) by multiplicative functionals M (not necessarily supermartingale), for instance, by Feynman-Kac functionals. Among others, this leads to an extension of the gauge theorem of Chung/Rao and even of one direction of the conditional gauge theorem of Falkner and Zhao.

MSC:

60J35 Transition functions, generators and resolvents
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Blanchard, Ph.; Ma, Zh.; Korezlioglu, H.; Ustunel, A. S., New results on the schrödinger semigroups with potentials given by signed smooth measures, Stochastic analysis and related topics II, 213-243 (1990), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0717.60075
[2] Boukricha, A.; Hansen, W.; Hueber, H., Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces, Expo. Math., 5, 97-135 (1987) · Zbl 0659.35025
[3] Chung, K. L., The gauge and conditional gauge theorem. Séminaire de Probabilités, 496-503 (1985), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0561.60084
[4] Chung, K. L.; Král, J.; Lukeš, J.; Netuka, I.; Veselý, J., Probability methods in potential theory, Potential theory—surveys and problems, 42-54 (1988), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0677.60085
[5] Chung, K. L.; Rao, M.; Çinlar, E.; Chung, K. L.; Getoor, R. K., Feynman-Kac functional and the Schrödinger equation, Seminar on stochastic processes 1981, 1-29 (1981), Basel: Birkhäuser, Basel · Zbl 0492.60073
[6] Chung, K. L.; Rao, M., General gauge theorem for multiplicative functionals, Trans. Am. Math. Soc., 306, 819-836 (1988) · Zbl 0647.60083
[7] Cranston, M.; Fabes, E.; Zhao, Z., Conditional gauge and potential theory for the Schrödinger operator, Trans. Am. Math. Soc., 307, 171-194 (1988) · Zbl 0652.60076
[8] Dellacherie, C.; Meyer, P. A., Probabilités et potentiel, vol. 1-4 (1976), Paris: Herrmann, Paris
[9] Falkner, N., Feynman—Kac functionals and positive solutions of 1/2Δu+qu=0.Z. Wahrscheinlichkeitstheor, Verw. Geb., 65, 19-33 (1983) · Zbl 0496.60078
[10] Feyel, D.; De La Pradelle, A., Étude de l’équation 1/2Δu−uμ=0 où μ est une mesure positive, Ann. Inst. Fourier, 38, 199-218 (1988) · Zbl 0645.35018
[11] Getoor, R. K., Transience and recurrence of Markov processes, Séminaire Probabilités, 397-409 (1980), Berlin Heidelberg New York: Springer, Berlin Heidelberg New York · Zbl 0431.60067
[12] Hansen, W.; Hueber, H., Eigenvalues in potential theory, J. Differ. Equation, 73, 133-152 (1988) · Zbl 0647.35062
[13] Hansen, W.; Ma, Zh., Perturbation by differences of unbounded potentials, Math. Ann., 287, 553-569 (1990) · Zbl 0685.31005
[14] Kröger, P., Harmonic spaces associated with parabolic and elliptic differential operators, Math. Ann., 285, 393-403 (1989) · Zbl 0664.31011
[15] Sharpe, M., General theory of Markov Processes (1988), Boston San Diego New York: Academic Press, Boston San Diego New York · Zbl 0649.60079
[16] Sturm, K.-Th., On the Dirichlet-poisson problem for Schrödinger operators, C.R. Math. Rep. Acad. Sci. Canada, IX, 149-154 (1987) · Zbl 0648.35024
[17] Sturm, K.-Th.; Král, J.; Lukeš, J.; Netuka, I.; Veselý, J., Schrödinger equations with discontinuous solutions, Potential theory, 333-337 (1988), New York: Plenum Press, New York · Zbl 0704.35041
[18] [St3] Sturm, K.-Th.: Störung von Hunt-Prozessen durch signierte additive Funktionale. Thesis, Erlangen 1989 · Zbl 0692.60057
[19] [St4] Sturm, K.-Th.: Measures changing no polar sets and additive functionals of Brownian motion. Forum Math. (to appear 1991)
[20] [St5] Sturm, K.-Th.: Schrödinger operators with highly singular, oscillating potentials. Preprint, Erlangen 1991
[21] Zhao, Z., Conditional gauge with unbounded potential, Z. Wahrscheinlichkeitstheor. Verw. Geb., 65, 13-18 (1983) · Zbl 0521.60074
[22] Zhao, Z., Uniform boundedness of conditional gauge and Schrödinger equations, Commun. Math. Phys., 93, 19-31 (1984) · Zbl 0545.35087
[23] Zhao, Z., Green function for Schrödinger operator and conditioned Feynman—Kac gauge, J. Math. Anal. Appl., 116, 309-334 (1986) · Zbl 0608.35012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.