Criticality and subcriticality of generalized Schrödinger forms with non-local perturbations

Li, Liping

doi:10.1090/proc/13523

Criticality and subcriticality of generalized Schrödinger forms with non-local perturbations
HTML articles powered by AMS MathViewer

by Liping Li PDF

Proc. Amer. Math. Soc. 145 (2017), 3929-3939 Request permission

Abstract:

In this paper, we shall treat the Schrödinger forms with non-local perturbations. We first extend the definitions of subcriticality, criticality and supercriticality for the Schrödinger forms by Takeda (2014) to the non-local cases in the context of quasi-regular Dirichlet forms. Then we prove an analytic characterization of these definitions via the bottom of the spectrum set.

References

Zhen-Qing Chen, Analytic characterization of conditional gaugeability for non-local Feynman-Kac transforms, J. Funct. Anal. 202 (2003), no. 1, 226–246. MR 1994771, DOI 10.1016/S0022-1236(02)00096-4
Zhen-Qing Chen, Gaugeability and conditional gaugeability, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4639–4679. MR 1926893, DOI 10.1090/S0002-9947-02-03059-3
Zhen-Qing Chen and Masatoshi Fukushima, Symmetric Markov processes, time change, and boundary theory, London Mathematical Society Monographs Series, vol. 35, Princeton University Press, Princeton, NJ, 2012. MR 2849840
Zhen-Qing Chen and Kazuhiro Kuwae, On doubly Feller property, Osaka J. Math. 46 (2009), no. 4, 909–930. MR 2604914
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, General gauge and conditional gauge theorems, Ann. Probab. 30 (2002), no. 3, 1313–1339. MR 1920109, DOI 10.1214/aop/1029867129
Zhen-Qing Chen and Renming Song, Conditional gauge theorem for non-local Feynman-Kac transforms, Probab. Theory Related Fields 125 (2003), no. 1, 45–72. MR 1952456, DOI 10.1007/s004400200219
Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, Second revised and extended edition, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 2011. MR 2778606
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
D. Kim and K. Kuwae, Analytic characterizations of gaugeability for generalized Feynman-Kac functionals, to appear in Trans. Amer. Math. Soc.
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
Zhi Ming Ma and Michael Röckner, Markov processes associated with positivity preserving coercive forms, Canad. J. Math. 47 (1995), no. 4, 817–840. MR 1346165, DOI 10.4153/CJM-1995-042-6
Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press, Inc., Boston, MA, 1988. MR 958914
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
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
Masayoshi Takeda, A variational formula for Dirichlet forms and existence of ground states, J. Funct. Anal. 266 (2014), no. 2, 660–675. MR 3132724, DOI 10.1016/j.jfa.2013.10.024
Masayoshi Takeda, Criticality and subcriticality of generalized Schrödinger forms, Illinois J. Math. 58 (2014), no. 1, 251–277. MR 3331850
Masayoshi Takeda and Toshihiro Uemura, Subcriticality and gaugeability for symmetric $\alpha$-stable processes, Forum Math. 16 (2004), no. 4, 505–517. MR 2044025, DOI 10.1515/form.2004.024
Jiangang Ying, Bivariate Revuz measures and the Feynman-Kac formula, Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), no. 2, 251–287. MR 1386221
Jiangang Ying, Dirichlet forms perturbated by additive functionals of extended Kato class, Osaka J. Math. 34 (1997), no. 4, 933–952. MR 1618693
Jiangang Ying, Revuz measures and related formulas on energy functional and capacity, Potential Anal. 8 (1998), no. 1, 1–19. MR 1608658, DOI 10.1023/A:1017966116127