Gaugeability and conditional gaugeability

Chen, Zhen-Qing

doi:10.1090/S0002-9947-02-03059-3

Gaugeability and conditional gaugeability
HTML articles powered by AMS MathViewer

by Zhen-Qing Chen PDF

Trans. Amer. Math. Soc. 354 (2002), 4639-4679 Request permission

Abstract:

New Kato classes are introduced for general transient Borel right processes, for which gauge and conditional gauge theorems hold. These new classes are the genuine extensions of the Green-tight measures in the classical Brownian motion case. However, the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability. We show that gaugeability, conditional gaugeability and the subcriticality for the associated Schrödinger operators are equivalent for transient Borel right processes with strong duals. Analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric Markov processes. These analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases. Connections with the positivity of the spectral radii of the associated Schrödinger operators are also established.

References

M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), no. 2, 209–273. MR 644024, DOI 10.1002/cpa.3160350206
A. B. Amor and W. Hansen, Continuity of eigenvalues for Schrödinger operators, $L^p$-properties of Kato type integral operators, Math. Ann. 321 (2001), 925-953.
Albert Benveniste and Jean Jacod, Systèmes de Lévy des processus de Markov, Invent. Math. 21 (1973), 183–198 (French). MR 343375, DOI 10.1007/BF01390195
R. M. Blumenthal and R. K. Getoor, Markov processes and potential theory, Pure and Applied Mathematics, Vol. 29, Academic Press, New York-London, 1968. MR 0264757
Zhen Qing Chen, Zhi Ming Ma, and Michael Röckner, Quasi-homeomorphisms of Dirichlet forms, Nagoya Math. J. 136 (1994), 1–15. MR 1309378, DOI 10.1017/S0027763000024934
Zhen-Qing Chen and Renming Song, Estimates on Green functions and Poisson kernels for symmetric stable processes, Math. Ann. 312 (1998), no. 3, 465–501. MR 1654824, DOI 10.1007/s002080050232
Zhen-Qing Chen and Renming Song, Intrinsic ultracontractivity and conditional gauge for symmetric stable processes, J. Funct. Anal. 150 (1997), no. 1, 204–239. MR 1473631, DOI 10.1006/jfan.1997.3104
Z.-Q. Chen and R. Song, General gauge and conditional gauge theorems. Preprint, 2000. To appear in Ann. Probab.
Z.-Q. Chen and R. Song, Conditional gauge theorem for non-local Feynman-Kac transforms. Preprint, 2001. To appear in Probab. Theory Related Fields.
Z.-Q. Chen and R. Song, Drift transforms and Green function estimates for discontinuous processes. Preprint, 2001.
K. L. Chung and K. M. Rao, Feynman-Kac functional and the Schrödinger equation, Seminar on Stochastic Processes, 1981 (Evanston, Ill., 1981) Progr. Prob. Statist., vol. 1, Birkhäuser, Boston, Mass., 1981, pp. 1–29. MR 647779
K. L. Chung and K. M. Rao, General gauge theorem for multiplicative functionals, Trans. Amer. Math. Soc. 306 (1988), no. 2, 819–836. MR 933320, DOI 10.1090/S0002-9947-1988-0933320-1
Kai Lai Chung and Zhong Xin Zhao, From Brownian motion to Schrödinger’s equation, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 312, Springer-Verlag, Berlin, 1995. MR 1329992, DOI 10.1007/978-3-642-57856-4
M. Cranston, E. Fabes, and Z. Zhao, Conditional gauge and potential theory for the Schrödinger operator, Trans. Amer. Math. Soc. 307 (1988), no. 1, 171–194. MR 936811, DOI 10.1090/S0002-9947-1988-0936811-2
Claude Dellacherie and Paul-André Meyer, Probabilités et potentiel. Chapitres V à VIII, Revised edition, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1385, Hermann, Paris, 1980 (French). Théorie des martingales. [Martingale theory]. MR 566768
C. Doléans-Dade, Quelques applications de la formule de changement de variables pour les semimartingales, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 16 (1970), 181–194 (French). MR 283883, DOI 10.1007/BF00534595
P. J. Fitzsimmons, On the excursions of Markov processes in classical duality, Probab. Theory Related Fields 75 (1987), no. 2, 159–178. MR 885460, DOI 10.1007/BF00354031
P. J. Fitzsimmons, Time changes of symmetric Markov processes and a Feynman-Kac formula, J. Theoret. Probab. 2 (1989), no. 4, 487–501. MR 1011201, DOI 10.1007/BF01051880
P. J. Fitzsimmons and R. K. Getoor, Revuz measures and time changes, Math. Z. 199 (1988), no. 2, 233–256. MR 958650, DOI 10.1007/BF01159654
P. J. Fitzsimmons and R. K. Getoor, Smooth measures and continuous additive functionals of right Markov processes, Itô’s stochastic calculus and probability theory, Springer, Tokyo, 1996, pp. 31–49. MR 1439516
Gerald B. Folland, Real analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. Modern techniques and their applications; A Wiley-Interscience Publication. MR 767633
Masatoshi Fukushima, Y\B{o}ichi Ōshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR 1303354, DOI 10.1515/9783110889741
R. K. Getoor, Transience and recurrence of Markov processes, Seminar on Probability, XIV (Paris, 1978/1979) Lecture Notes in Math., vol. 784, Springer, Berlin, 1980, pp. 397–409. MR 580144
R. K. Getoor, Measure perturbations of Markovian semigroups, Potential Anal. 11 (1999), no. 2, 101–133. MR 1703827, DOI 10.1023/A:1008615732680
R. K. Getoor and J. Glover, Riesz decompositions in Markov process theory, Trans. Amer. Math. Soc. 285 (1984), no. 1, 107–132. MR 748833, DOI 10.1090/S0002-9947-1984-0748833-1
Tadeusz Kulczycki, Properties of Green function of symmetric stable processes, Probab. Math. Statist. 17 (1997), no. 2, Acta Univ. Wratislav. No. 2029, 339–364. MR 1490808
Hiroshi Kunita and Takesi Watanabe, Notes on transformations of Markov processes connected with multiplicative functionals, Mem. Fac. Sci. Kyushu Univ. Ser. A 17 (1963), 181–191. MR 163358, DOI 10.2206/kyushumfs.17.181
Zhi Ming Ma and Michael Röckner, Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992. MR 1214375, DOI 10.1007/978-3-642-77739-4
P. A. Meyer, Note sur l’interprétation des mesures d’équilibre, Séminaire de Probabilités, VII (Univ. Strasbourg, année universitaire 1971–1972), Lecture Notes in Math., Vol. 321, Springer, Berlin, 1973, pp. 210–216 (French). MR 0373030
Yehuda Pinchover, Criticality and ground states for second-order elliptic equations, J. Differential Equations 80 (1989), no. 2, 237–250. MR 1011149, DOI 10.1016/0022-0396(89)90083-1
D. Revuz, Mesures associées aux fonctionnelles additives de Markov. I, Trans. Amer. Math. Soc. 148 (1970), 501–531 (French). MR 279890, DOI 10.1090/S0002-9947-1970-0279890-7
Sadao Sato, An inequality for the spectral radius of Markov processes, Kodai Math. J. 8 (1985), no. 1, 5–13. MR 776702, DOI 10.2996/kmj/1138036992
Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press, Inc., Boston, MA, 1988. MR 958914
Martin L. Silverstein, The sector condition implies that semipolar sets are quasi-polar, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 41 (1977/78), no. 1, 13–33. MR 467934, DOI 10.1007/BF00535011
Barry Simon, Schrödinger semigroups, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 3, 447–526. MR 670130, DOI 10.1090/S0273-0979-1982-15041-8
Peter Stollmann and Jürgen Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), no. 2, 109–138. MR 1378151, DOI 10.1007/BF00396775
W. Stummer and K.-Th. Sturm, On exponentials of additive functionals of Markov processes, Stochastic Process. Appl. 85 (2000), no. 1, 45–60. MR 1730619, DOI 10.1016/S0304-4149(99)00064-2
Karl-Theodor Sturm, Gauge theorems for resolvents with application to Markov processes, Probab. Theory Related Fields 89 (1991), no. 4, 387–406. MR 1118555, DOI 10.1007/BF01199785
Masayoshi Takeda, Exponential decay of lifetimes and a theorem of Kac on total occupation times, Potential Anal. 11 (1999), no. 3, 235–247. MR 1717103, DOI 10.1023/A:1008649623291
M. Takeda, Conditional gaugeability and subcriticality of generalized Schrödinger operators. Preprint, 2001. To appear in J. Funct. Anal.
Jiangang Ying, Dirichlet forms perturbated by additive functionals of extended Kato class, Osaka J. Math. 34 (1997), no. 4, 933–952. MR 1618693
Z. Zhao, A probabilistic principle and generalized Schrödinger perturbation, J. Funct. Anal. 101 (1991), no. 1, 162–176. MR 1132313, DOI 10.1016/0022-1236(91)90153-V
Z. Zhao, Subcriticality and gaugeability of the Schrödinger operator, Trans. Amer. Math. Soc. 334 (1992), no. 1, 75–96. MR 1068934, DOI 10.1090/S0002-9947-1992-1068934-5