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On the Upper Rate Functions of Some Time Inhomogeneous Diffusion Processes

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Abstract

In the present paper, we study an upper escape rate of some time inhomogeneous diffusion process associated with a family of regular and local Dirichlet forms. In particular, by making full use of Gaussian type’s heat kernel estimates, we establish integral tests for an upper rate function of the time inhomogeneous diffusion process with a coefficient that is not necessarily bounded concerning space and time.

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References

  1. Barlow, M.T., Grigor’yan, A., Kumagai, T.: On the equivalence of parabolic Harnack inequalities and heat kernel estimates, J. Math. Soc. Japan, no.4, 64, 1091–1146 (2012)

  2. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, Second revised and extended edition, de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin. 19, (2011)

  3. Grigor’yan, A.: Escape rate of Brownian motion on Riemannian manifolds, Appl. Anal. no.1-4, 71, 63–89 (1999)

  4. Grigor’yan, A., Hsu, E.: Volume growth and escape rate of Brownian motion on a Cartan-Hadamard manifold, Sobolev spaces in mathematics. II, 209 225, International Mathematical Series, Springer, New York, (2009)

  5. Grigor’yan, A., Kelbert, M.: Range of fluctuation of Brownain motion on a complete Riemannian manifold, Ann. Probab. no.1, 26, 78–111 (1998)

  6. Hsu, E., Qin, G.: Volume growth and escape rate of Brownian motion on a complete Riemannian manifold, Ann. Probab., no.4, 38, 1570–1582 (2010)

  7. Huang, X.: Escape rate of Markov chains on infinite graphs, J. Theoret. Probab., no.2, 27, 634–682 (2014)

  8. Huang, X., Shiozawa, Y.: Upper escape rate of Markov chains on weighted graphs, Stoch. Process. Appl., no.1, 124, 317–347 (2014)

  9. Ito, K., McKean, H.P., Jr.: Diffusion processes and their sample paths, Die Grundlehren der Mathematischen Wissenschaften, vol. 125. Academic Press Inc, Publishers, New York, Springer-Verlag, Berlin-New York (1965)

    Google Scholar 

  10. Lierl, J.: Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms. J. Math. Pures. Appl. 140, 1–66 (2020)

    Article  MathSciNet  Google Scholar 

  11. Moser, J.: A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17, 101–134 (1964)

    Article  MathSciNet  Google Scholar 

  12. Oshima, Y.: Semi-Dirichlet forms and Markov processes, De Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin, 48 (2013)

  13. Ouyang, S.: Volume growth, comparison theorem and escape rate of diffusion processes, Stochastics, no.3, 88, 353–372 (2016)

  14. Saloff-Coste, L.: Aspects of Sobolev-Type Inequalites, London Mathematical Society Lecture Note Series, 289 (2001)

  15. Saloff-Coste, L., Stroock, D.: Opérateurs uniformément sous-elliptiques sur les groupes de Lie, J. Funct. Anal., no.1, 98, 97–121 (1991)

  16. Shiozawa, Y.: Escape rate of symmetric jump-diffusion processes, Trans. Amer. Math. Soc., no.11, 368, 7645–7680 (2016)

  17. Shiozawa, Y., Wang, J.: Rate functions for symmetric Markov processes via heat kernel, Potential Anal., no.1, 46, 23–53 (2017)

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

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

  20. Yan, J.A.: A formula for densities of transition functions, Séminaire de Probabilités, XXII, Lecture Notes in Math., Springer, Berlin, 1321, 92–100 (1988)

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Acknowledgements

The authors would like to thank the referee for his/her valuable comments and suggestions. The first named author is partially supported by JSPS KAKENHI Grant number 20K03635.

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Japan Society for the Promotion of Science, 20K03635.

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Authors and Affiliations

  1. Faculty of Advanced Science and Technology, Kumamoto University, Kumamoto, 860-8555, Japan

    Daehong Kim

  2. Department of Mathematics and Engineering, Kumamoto University, Kumamoto, 860-8555, Japan

    Yoichi Oshima

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  1. Daehong Kim
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  2. Yoichi Oshima
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Daehong Kim and Yoichi Oshima wrote and reviewed the manuscript.

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Correspondence to Daehong Kim.

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Kim, D., Oshima, Y. On the Upper Rate Functions of Some Time Inhomogeneous Diffusion Processes. Potential Anal 60, 1181–1213 (2024). https://doi.org/10.1007/s11118-023-10083-8

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  • DOI: https://doi.org/10.1007/s11118-023-10083-8

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