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Harnack Inequality and Applications for Infinite-Dimensional GEM Processes

Abstract

The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in Feng and Wang (J. Appl. Probab. 44 938–949 2007) to simulate the two-parameter GEM distributions. In particular, the associated Dirichlet form satisfies the super log-Sobolev inequality which strengthens the log-Sobolev inequality derived in Feng and Wang (J. Appl. Probab. 44 938–949 2007). To prove the main results, explicit Harnack inequality and super Poincaré inequality are established for the one-dimensional Wright-Fisher diffusion processes. The main tool of the study is the coupling by change of measures.

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References

  1. 1.

    Arnaudon, M., Thalmaier, A., Wang, F.-Y.: Harnack inequality heat kernel estimates on manifolds with curvature unbounded below. Bull. Sci. Math. 130, 223–233 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. 2.

    Arnaudon, M., Thalmaier, A., Wang, F.-Y.: Gradient estimates Harnack inequalities on non-compact Riemannian manifolds. Stoch. Proc. Appl. 119, 3653–3670 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  3. 3.

    Connor, R.J., Moismann, J.E.: Concepts of independence for proportions with a generalization of the Dirichlet distribution. J. Amer. Statist. Assoc. 64, 194–206 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  4. 4.

    Coulhon, T., Kerkyacharian, G., Petrushev, P.: Heat kernel generated frames in the setting of Dirichlet spaces. J. Fourier Anal. Appl. 18, 995–1066 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. 5.

    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  6. 6.

    Epstein, C.L., Mazzeo, R.: Wright-Fisher diffusion in one dimension. SIAM J. Math. Anal. 42, 568–608 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  7. 7.

    W.J: Ewens, vol. I. Springer-Verlag, New York (2004)

  8. 8.

    Feng, S., Wang, F.-Y.: A class of infinite-dimensional diffusion processes with connection to population genetics. J. Appl. Probab. 44, 938–949 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. 9.

    Feng, S., Sun, W., Wang, F.-Y., Xu, F.: Functional inequalities for the two-parameter extension of the infinite-many-neutral-alles diffusion. J. Funct. Anal. 260, 399–413 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. 10.

    Gong, F.-Z., Wang, F.-Y.: Heat kernel estimates with application to compactness of manifolds. Quart. J. Math. 52, 171–180 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. 11.

    Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd Ed. North-Holland, Amsterdam (1989)

    MATH  Google Scholar 

  12. 12.

    Ishwaran, H., James, L.F.: Gibbs sampling methods for stick-breaking priors. J. Amer. Statist. Assoc. 96, 161–173 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. 13.

    Wang, F.-Y.: Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab. Theory Relat. Fields 109, 417–424 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. 14.

    Wang, F.-Y.: Functional inequalities for empty essential spectrum. J. Funct. Anal. 170, 219–245 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. 15.

    Wang, F.-Y.: Functional inequalities, semigroup properties and spectrum estimates, Infinite Dimensional Analysis. Quantum Probab. Relat. Top. 3, 263–295 (2000)

    MATH  Google Scholar 

  16. 16.

    Wang, F.-Y: Functional Inequality, Markov Semigroups, and Spectral Theory. Science Press, Beijing (2005)

    Google Scholar 

  17. 17.

    Wang, F.-Y.: Harnack Inequalities and Applications for Stochastic Partial Differential Equations. Springer, Berlin (2013)

    Book  Google Scholar 

  18. 18.

    Wang, F.-Y., Yuan, C.: Harnack inequalities for functional SDEs with multiplicative noise and applications. Stoch. Proc. Appl. 121, 2692–2710 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Correspondence to Feng-Yu Wang.

Additional information

Supported in part by NNSFC(11131003, 11431014), the 985 project, the Laboratory of Mathematical and Complex Systems, and NSERC.

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Feng, S., Wang, FY. Harnack Inequality and Applications for Infinite-Dimensional GEM Processes. Potential Anal 44, 137–153 (2016). https://doi.org/10.1007/s11118-015-9502-5

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Keywords

  • GEM distribution
  • GEM diffusion process
  • Harnack inequality
  • Heat kernel
  • Super log-Sobolev inequality

Mathematics Subject Classification (2010)

  • 65G17
  • 65G60