Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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


Author: Liping Li
Journal: Proc. Amer. Math. Soc. 145 (2017), 3929-3939
MSC (2010): Primary 31C25, 31C05, 60J25
DOI: https://doi.org/10.1090/proc/13523
Published electronically: February 15, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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, https://doi.org/10.1016/S0022-1236(02)00096-4
  • [2] Zhen-Qing Chen, Gaugeability and conditional gaugeability, Trans. Amer. Math. Soc. 354 (2002), no. 11, 4639-4679 (electronic). MR 1926893, https://doi.org/10.1090/S0002-9947-02-03059-3
  • [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
  • [4] Zhen-Qing Chen and Kazuhiro Kuwae, On doubly Feller property, Osaka J. Math. 46 (2009), no. 4, 909-930. MR 2604914
  • [5] Zhen Qing Chen, Zhi Ming Ma, and Michael Röckner, Quasi-homeomorphisms of Dirichlet forms, Nagoya Math. J. 136 (1994), 1-15. MR 1309378
  • [6] Zhen-Qing Chen and Renming Song, General gauge and conditional gauge theorems, Ann. Probab. 30 (2002), no. 3, 1313-1339. MR 1920109, https://doi.org/10.1214/aop/1029867129
  • [7] 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, https://doi.org/10.1007/s004400200219
  • [8] 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
  • [9] 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
  • [10] D. Kim and K. Kuwae, Analytic characterizations of gaugeability for generalized Feynman-Kac functionals, to appear in Trans. Amer. Math. Soc.
  • [11] Zhi Ming Ma and Michael Röckner, Introduction to the theory of (nonsymmetric) Dirichlet forms, Universitext, Springer-Verlag, Berlin, 1992. MR 1214375
  • [12] 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, https://doi.org/10.4153/CJM-1995-042-6
  • [13] Michael Sharpe, General theory of Markov processes, Pure and Applied Mathematics, vol. 133, Academic Press, Inc., Boston, MA, 1988. MR 958914
  • [14] 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, https://doi.org/10.1016/S0304-4149(99)00064-2
  • [15] Peter Stollmann and Jürgen Voigt, Perturbation of Dirichlet forms by measures, Potential Anal. 5 (1996), no. 2, 109-138. MR 1378151, https://doi.org/10.1007/BF00396775
  • [16] Masayoshi Takeda, A variational formula for Dirichlet forms and existence of ground states, J. Funct. Anal. 266 (2014), no. 2, 660-675. MR 3132724, https://doi.org/10.1016/j.jfa.2013.10.024
  • [17] Masayoshi Takeda, Criticality and subcriticality of generalized Schrödinger forms, Illinois J. Math. 58 (2014), no. 1, 251-277. MR 3331850
  • [18] Masayoshi Takeda and Toshihiro Uemura, Subcriticality and gaugeability for symmetric $ \alpha $-stable processes, Forum Math. 16 (2004), no. 4, 505-517. MR 2044025, https://doi.org/10.1515/form.2004.024
  • [19] Jiangang Ying, Bivariate Revuz measures and the Feynman-Kac formula, Ann. Inst. H. Poincaré Probab. Statist. 32 (1996), no. 2, 251-287. MR 1386221
  • [20] Jiangang Ying, Dirichlet forms perturbated by additive functionals of extended Kato class, Osaka J. Math. 34 (1997), no. 4, 933-952. MR 1618693
  • [21] Jiangang Ying, Revuz measures and related formulas on energy functional and capacity, Potential Anal. 8 (1998), no. 1, 1-19. MR 1608658, https://doi.org/10.1023/A:1017966116127

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 31C25, 31C05, 60J25

Retrieve articles in all journals with MSC (2010): 31C25, 31C05, 60J25


Additional Information

Liping Li
Affiliation: Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
Email: liping_li@amss.ac.cn

DOI: https://doi.org/10.1090/proc/13523
Received by editor(s): March 23, 2016
Received by editor(s) in revised form: September 20, 2016, and September 27, 2016
Published electronically: February 15, 2017
Additional Notes: This research was partially supported by a joint grant (No. 2015LH0043) of China Postdoctoral Science Foundation and Chinese Academy of Science, and China Postdoctoral Science Foundation (No. 2016M590145)
Communicated by: David Levin
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society