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Potential Analysis

, Volume 48, Issue 3, pp 257–300 | Cite as

An Integrated Version of Varadhan’s Asymptotics for Lower-Order Perturbations of Strong Local Dirichlet Forms

  • Masanori Hino
  • Kouhei Matsuura
Open Access
Article
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Abstract

The studies of J. A. Ramírez, Hino–Ramírez, and Ariyoshi–Hino showed that an integrated version of Varadhan’s asymptotics holds for Markovian semigroups associated with arbitrary strong local symmetric Dirichlet forms. In this paper, we consider non-symmetric bilinear forms that are the sum of strong local symmetric Dirichlet forms and lower-order perturbed terms. We give sufficient conditions for the associated semigroups to have asymptotics of the same type.

Keywords

Varadhan’s asymptotics Short-time behavior Dirichlet form Intrinsic distance 

Mathematics Subject Classification (2010)

31C25 60J60 58J37 47D07 

Notes

Acknowledgements

This study was supported by JSPS KAKENHI Grant Number JP15H03625.

References

  1. 1.
    Ariyoshi, T., Hino, M.: Small-time asymptotic estimate in local Dirichlet spaces. Electron. J. Probab. 10, 1236–1259 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bauer, H.: Measure and integration theory. de Gruyter Studies in Mathematics, vol. 26. Walter de Gruyter, Berlin (2001)Google Scholar
  3. 3.
    Bouleau, N., Hirsch, F.: Dirichlet forms and analysis on Wiener Space. de Gruyter Studies in Mathematics, vol. 14. Walter de Gruyter, Berlin (1991)Google Scholar
  4. 4.
    Davies, E.B.: Heat kernels and spectral theory. Cambridge Tracts in Mathematics, vol. 92. Cambridge University Press, Cambridge (1990)Google Scholar
  5. 5.
    Fitzsimmons, P.J., Kuwae, K.: Non-symmetric perturbations of symmetric Dirichlet forms. J. Funct. Anal. 208, 140–162 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet forms and symmetric Markov processes, 2nd edn. de Gruyter Studies in Mathematics, vol. 19. Walter de Gruyter, Berlin (2011)Google Scholar
  7. 7.
    Hino, M.: Existence of invariant measures for diffusion processes on a Wiener space. Osaka J. Math. 35, 717–734 (1998)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Hino, M.: On short time asymptotic behavior of some symmetric diffusions on general state spaces. Potential Anal. 16, 249–264 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hino, M., Ramírez, J.A.: Small-time Gaussian behavior of symmetric diffusion semigroups. Ann. Probab. 14, 1254–1295 (2003)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Kuwae, K.: Invariant sets and ergodic decomposition of local semi-Dirichlet forms. Forum Math. 23, 1259–1279 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Liskevich, V.: On C 0-semigroups generated by elliptic second order differential expressions on L p-spaces. Differ. Integral Equ. 9, 811–826 (1996)zbMATHGoogle Scholar
  12. 12.
    Lunt, J., Lyons, T.J., Zhang, T.S.: Integrability of functionals of Dirichlet processes, probabilistic representations of semigroups, and estimates of heat kernels. J. Funct. Anal. 153, 320–342 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ma, Z.M., Röckner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  14. 14.
    Ma, Z.M., Röckner, M.: Markov processes associated with positivity preserving coercive forms. Canad. J. Math. 47, 817–840 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Norris, J.R.: Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. Acta Math. 179, 79–103 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ramírez, J.A.: Short time asymptotics in Dirichlet spaces. Comm. Pure Appl. Math. 54, 259–293 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Röckner, M., Zhang, T.S.: Probabilistic representations and hyperbound estimates for semigroups. Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 2, 337–358 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Sobol, Z., Vogt, H.: On the L p-theory of C 0-semigroups associated with second order elliptic operators. I. J. Funct. Anal. 193, 24–54 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Varadhan, R.: On the behavior of the fundamental solution of the heat equation with variable coefficients. Comm. Pure Appl. Math. 20, 431–455 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Voigt, J.: One-parameter semigroups acting simultaneously on different L p spaces. Bull. Soc. Roy. Sci. Liège 61, 465–470 (1992)MathSciNetzbMATHGoogle Scholar

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsKyoto UniversityKyotoJapan
  2. 2.Mathematical InstituteTohoku UniversityAobaJapan

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