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Construction of Generalized Diffusion Processes: the Resolvent Approach

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Abstract

In this paper, we define the generalized diffusion operator \(L = {d \over {dM}}{d \over {dS}}\) for two suitable measures on the line, which includes the generators of the birth-death processes, the one-dimensional diffusion and the gap diffusion among others. Via the standard resolvent approach, the associated generalized diffusion processes are constructed.

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References

  1. Bogachev, V. I.: Measure Theory I, Springer-Verlag, Berlin, 2007

    Book  Google Scholar 

  2. Chen, M. F.: From Markov Chains to Non-equilibrium Particle Systems, Second edition, World Scientific, 2004

  3. Chen, M. F.: Eigenvalues, Inequalities, and Ergodic Theory, Springer-Verlag, London, 2005

    MATH  Google Scholar 

  4. Feller, W.: The parabolic differential equations and the associated semi-groups of transformations. Ann. of Math., 55, 468–519 (1952)

    Article  MathSciNet  Google Scholar 

  5. Feller, W.: Diffusion processes in one dimension. Trans. Amer. Math. Soc., 77, 1–31 (1954)

    Article  MathSciNet  Google Scholar 

  6. Feller, W.: Generalized second order differential operators and their lateral conditions. Illinois J. Math., 1, 459–504 (1957)

    Article  MathSciNet  Google Scholar 

  7. Feller, W.: The birth and death processes as diffusion processes. J. Math. Pures Appl., 38, 301–345 (1959)

    MathSciNet  MATH  Google Scholar 

  8. Itô, K.: Stochastic Processes, Matematisk Institut, Aarhus, 1969

    MATH  Google Scholar 

  9. Itô, K., Mckean, H. P.: Diffusion Processes and Their Sample Paths, Second printing, Springer-Verlag, Berlin-New York, 1974

    MATH  Google Scholar 

  10. Kigami, J., Lapidus, M. L.: Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals. Comm. Math. Phys., 158, 93–125 (1993)

    Article  MathSciNet  Google Scholar 

  11. Kotani, S., Watanabe, S.: Krein’s spectral theory of strings and generalized diffusion processes. In: Functional Analysis in Markov Processes, (Katata/Kyoto, 1981), pp. 235–259, Lecture Notes in Math., Vol. 923, Springer, Berlin-New York, 1982

    Chapter  Google Scholar 

  12. Li, Y., Mao, Y. H.: The optimal constant in generalized Hardy’s inequality. Math. Inequal. Appl., to appear, (2020)

  13. Qian, M. P., Gong, G. L., Qian, M.: Reversibility of the minimal Markov process generated by a second-order differential operator (Chinese). Beijing Daxue Xuebao, 2, 19–45 (1979)

    Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Computational Science, Xiangtan University, Xiangtan, 411105, P. R. China

    Ying Li

  2. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, P. R. China

    Yong Hua Mao

Authors
  1. Ying Li
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  2. Yong Hua Mao
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Correspondence to Yong Hua Mao.

Additional information

Supported in part by NSFC (Grant No. 11771047) and Hu Xiang Gao Ceng Ci Ren Cai Ju Jiao Gong Cheng-Chuang Xin Ren Cai (Grant No. 2019RS1057)

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Li, Y., Mao, Y.H. Construction of Generalized Diffusion Processes: the Resolvent Approach. Acta. Math. Sin.-English Ser. 36, 691–710 (2020). https://doi.org/10.1007/s10114-020-9282-8

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  • DOI: https://doi.org/10.1007/s10114-020-9282-8

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