Skip to main content
SpringerLink
Log in

Recurrence Criteria for Generalized Dirichlet Forms

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

We develop sufficient analytic conditions for recurrence and transience of non-sectorial perturbations of possibly non-symmetric Dirichlet forms on a general state space. These form an important subclass of generalized Dirichlet forms which were introduced in Stannat (Ann Scuola Norm Sup Pisa Cl Sci (4) 28(1):99–140, 1999). In case there exists an associated process, we show how the analytic conditions imply recurrence and transience in the classical probabilistic sense. As an application, we consider a generalized Dirichlet form given on a closed or open subset of \(\mathbb {R}^d\) which is given as a divergence free first-order perturbation of a non-symmetric energy form. Then, using volume growth conditions of the sectorial and non-sectorial first-order part, we derive an explicit criterion for recurrence. Moreover, we present concrete examples with applications to Muckenhoupt weights and counterexamples. The counterexamples show that the non-sectorial case differs qualitatively from the symmetric or non-symmetric sectorial case. Namely, we make the observation that one of the main criteria for recurrence in these cases fails to be true for generalized Dirichlet forms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bhattacharya, R.N.: Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Probab. 6(4), 541–553 (1978)

    Article  MathSciNet  Google Scholar 

  2. Beznea, L., Cîmpean, I., Röckner, M.: Irreducible recurrence, ergodicity, and extremality of invariant measures for resolvents. arXiv:1409.6492

  3. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin (2011)

    MATH  Google Scholar 

  4. Fukushima, M.: Transience, Recurrence and Large Deviation of Markov Processes. Bielefeld IGK Seminar (2007)

  5. Gim M., Trutnau G.: Explicit Recurrence Criteria for Symmetric Gradient Type Dirichlet Forms Satisfying a Hamza Type Condition, vol. 15(65), No. 4. Mathematical reports (2013)

  6. Getoor R.K.: Transience and Recurrence of Markov Processes. Seminar on Probability, XIV. Lecture Notes in Mathematics, vol. 784, pp. 397–409. Springer, Berlin (1980)

    Google Scholar 

  7. Khas’minskii, R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Theory Probab. Appl. 5(2), 179–196 (1960)

    Article  MathSciNet  Google Scholar 

  8. Kuwae, K.: Invariant sets and ergodic decomposition of local semi-Dirichlet forms. Forum Math. 23(6), 1259–1279 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Ma, Z., Röckner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)

    Book  Google Scholar 

  10. Masamune, J., Uemura, T., Wang, J.: On the conservativeness and the recurrence of symmetric jump-diffusions. J. Funct. Anal. 263(12), 3984–4008 (2012)

    Article  MathSciNet  Google Scholar 

  11. Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer, London (1993)

    Book  Google Scholar 

  12. Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  13. Oshima, Y.: On conservativeness and recurrence criteria for Markov processes. Potential Anal. 1(2), 115–131 (1992)

    Article  MathSciNet  Google Scholar 

  14. Oshima, Y.: Time-dependent Dirichlet forms and related stochastic calculus. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 7(2), 281–316 (2004)

    Article  MathSciNet  Google Scholar 

  15. Oshima, Y.: Semi-Dirichlet Forms and Markov Processes. Walter de Gruyter, Berlin (2013)

    Book  Google Scholar 

  16. Ouknine, Y., Russo, F., Trutnau, G.: On countably skewed Brownian motion with accumulation point. Electron. J. Probab. 20(82), 1–27 (2015)

    MathSciNet  MATH  Google Scholar 

  17. Pinsky, R.G.: Positive Harmonic Functions and Diffusion. Cambridge Studies in Advanced Mathematics, vol. 45. Cambridge University Press, Cambridge (1995)

    Book  Google Scholar 

  18. Da Prato, G., Röckner, M.: Singular dissipative stochastic equations in Hilbert spaces. Probab. Theory Relat. Fields 124(2), 261–303 (2002)

    Article  MathSciNet  Google Scholar 

  19. Röckner, M., Shin, J., Trutnau, G.: Non symmetric distorted Brownian motion: strong solutions, strong Feller property and non-explosion results. Discrete Contin. Dyn. Syst. Ser. B 21(9), 3219–3237 (2016)

    Article  MathSciNet  Google Scholar 

  20. Schilling, R.L.: A note on invariant sets. Probab. Math. Statics 24(1), 47–66 (2004)

    MathSciNet  MATH  Google Scholar 

  21. Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and \(L^p\)- Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)

    MathSciNet  MATH  Google Scholar 

  22. Sturm, K.T.: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32(2), 275–312 (1995)

    MathSciNet  MATH  Google Scholar 

  23. Sturm, K.T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75(3), 273–297 (1996)

    MathSciNet  MATH  Google Scholar 

  24. Stannat, W.: (Nonsymmetric) Dirichlet operators on \(L^1\): existence, uniqueness and associated Markov processes. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28(1), 99–140 (1999)

    MathSciNet  MATH  Google Scholar 

  25. Stannat W.: The Theory of Generalized Dirichlet Forms and Its Applications in Analysis and Stochastics. Dissertation, Bielefeld 1996. Published as Memoirs of the AMS, vol. 142. No. 678 (1999)

  26. Torchinsky, A.: Real-Variable Methods in Harmonic Analysis. Courier Corporation, North Chelmsford (2012)

    MATH  Google Scholar 

  27. Trutnau, G.: Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non reflection part. Probab. Theory Relat. Fields 127(4), 455–495 (2003)

    Article  MathSciNet  Google Scholar 

  28. Turesson B.O.: Nonlinear Potential Theory and Weighted Sobolev Spaces. Lecture Notes in Mathematics, vol. 1736. Springer (2000)

Download references

Acknowledgements

The second named author would like to thank Wilhelm Stannat for leaving him his notes on recurrence for personal use several years ago. The research of Minjung Gim was supported by Global Ph.D. Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2012-H1A2A1004352). The research of Gerald Trutnau was supported by NRF-DFG Collaborative Research program and Basic Science Research Program through the National Research Foundation of Korea (NRF-2012K2A5A6047864 and NRF-2012R1A1A2006987) and by DFG through Grant Ro 1195/10-1.

Author information

Authors and Affiliations

  1. National Institute for Mathematical Sciences, 70 Yuseong-daero 1689 beon-gil, Yuseong-gu, Daejeon, 34047, South Korea

    Minjung Gim

  2. Department of Mathematical Sciences, Research Institute of Mathematics, Seoul National University, 599 Gwanak-Ro, Gwanak-Gu, Seoul, 08826, South Korea

    Gerald Trutnau

Authors
  1. Minjung Gim
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerald Trutnau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerald Trutnau.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gim, M., Trutnau, G. Recurrence Criteria for Generalized Dirichlet Forms. J Theor Probab 31, 2129–2166 (2018). https://doi.org/10.1007/s10959-017-0779-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-017-0779-8

Keywords

Mathematics Subject Classification (2010)

Search

Navigation