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On notions of harmonicity for non-symmetric Dirichlet form


In this paper, we extend the equivalence of the analytic and probabilistic notions of harmonicity in the context of Hunt processes associated with non-symmetric Dirichlet forms on locally compact separable metric spaces. Extensions to the processes associated with semi-Dirichlet forms and nearly symmetric right processes on Lusin spaces including infinite dimensional spaces are mentioned at the end of this paper.

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  1. 1

    Chen Z Q. On notions of harmonicity. Proc Amer Math Soc, 2009, 137: 3497–3510

    MATH  Article  MathSciNet  Google Scholar 

  2. 2

    Chen Z Q, Fukushima M. One-point extension of Markov processes by darning. Probab Theory Related Fields, 2008, 141: 61–112

    MATH  Article  MathSciNet  Google Scholar 

  3. 3

    Chen Z Q, Fukushima M. Symmetric Markov Processes, Time Change and Boundary Theory. Book manuscript, 2009

  4. 4

    Chen ZQ, Ma ZM, Röckner M. Quasi-homeomorphisms of Dirichlet forms. Nagoya Math J, 1994, 136: 1–15

    MATH  MathSciNet  Google Scholar 

  5. 5

    Chen C Z, Sun W. Perturbation of non-symmetric Dirichlet forms and associated Markov processes. Acta Math Sci, 2001, 21: 145–153

    MATH  MathSciNet  Google Scholar 

  6. 6

    Chen Z Q, Zhao Z. Diffusion processes and second order elliptic operators with singular coefficients for lower order terms. Math Ann, 1995, 302: 323–357

    MATH  Article  MathSciNet  Google Scholar 

  7. 7

    Fitzsimmons P J. On the quasi-regularity of semi-Dirichlet forms. Potential Analysis, 2001, 15: 151–185

    MATH  Article  MathSciNet  Google Scholar 

  8. 8

    Fukushima M, Oshima Y, Takeda M. Dirichlet Forms and Symmetric Markov Processes. Berlin: Walter de Gruyter, 1994

    MATH  Google Scholar 

  9. 9

    Hu Z C. Beurling-Deny formula of non-symmetric Dirichlet forms and the theory of semi-Dirichlet forms. PhD Dissertation. Chengdu: Sichuan University, 2004

    Google Scholar 

  10. 10

    Hu Z C, Ma Z M. Beurling-Deny formula of semi-symmetric Dirichlet forms. C R Math Acad Sci Paris, 2004, 338: 521–526

    MATH  MathSciNet  Google Scholar 

  11. 11

    Hu Z C, Ma Z M, Sun W. Extensions of Levy-Khintchine formula and Beurling-Deny formula in semi-Dirichlet forms setting. J Funct Anal, 2006, 239: 179–213

    MATH  Article  MathSciNet  Google Scholar 

  12. 12

    Hu Z C, Ma Z M, Sun W. On representations of non-symmetric Dirichlet forms. Potential Analysis, doi 10.1007/s11118-009-9145-5, 2009

  13. 13

    Ma Z M, Overbeck L, Röckner M. Markov processes associated with semi-Dirichlet forms. Osaka J Math, 1995, 32: 97–119

    MATH  MathSciNet  Google Scholar 

  14. 14

    Ma Z M, Röckner M. Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Berlin-Heidelberg-New York: Springer-Verlag, 1992

    MATH  Google Scholar 

  15. 15

    Oshima Y. Lectures on Dirichlet forms. Preprint, 1988

  16. 16

    Sato K. Lévy processes and infinitely divisible distributions. Cambridge: Cambridge University Press, 1999

    MATH  Google Scholar 

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Correspondence to RongChan Zhu.

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Ma, Z., Zhu, R. & Zhu, X. On notions of harmonicity for non-symmetric Dirichlet form. Sci. China Math. 53, 1407–1420 (2010).

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  • harmonic functions
  • uniformly integrable martingale
  • SPV integrable
  • non-symmetric Dirichlet forms
  • non-symmetric Beurling-Deny decomposition
  • Hunt processes


  • 30P12
  • 32C12