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Acta Applicandae Mathematicae

, Volume 146, Issue 1, pp 113–143 | Cite as

Global Dirichlet Heat Kernel Estimates for Symmetric Lévy Processes in Half-Space

  • Zhen-Qing Chen
  • Panki Kim
Article

Abstract

In this paper, we derive explicit sharp two-sided estimates for the Dirichlet heat kernels of a large class of symmetric (but not necessarily rotationally symmetric) Lévy processes on half spaces for all \(t>0\). These Lévy processes may or may not have Gaussian component. When Lévy density is comparable to a decreasing function with damping exponent \(\beta\), our estimate is explicit in terms of the distance to the boundary, the Lévy exponent and the damping exponent \(\beta\) of Lévy density.

Keywords

Dirichlet heat kernel Transition density Survival probability Exit time Lévy system Lévy process Symmetric Lévy process 

Mathematics Subject Classification (2000)

60J35 47G20 60J75 47D07 

Notes

Acknowledgement

We thank the referees for helpful comments.

References

  1. 1.
    Barlow, M.T., Bass, R.F., Chen, Z.-Q., Kassmann, M.: Non-local Dirichlet forms and symmetric jump processes. Trans. Am. Math. Soc. 361, 1963–1999 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bogdan, K., Grzywny, T.: Heat kernel of fractional Laplacian in cones. Colloq. Math. 118, 365–377 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bogdan, K., Grzywny, T., Ryznar, M.: Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Ann. Probab. 38, 1901–1923 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bogdan, K., Grzywny, T., Ryznar, M.: Density and tails of unimodal convolution semigroups. J. Funct. Anal. 266, 3543–3571 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bogdan, K., Grzywny, T., Ryznar, M.: Barriers, exit time and survival probability for unimodal Lévy processes. Probab. Theory Relat. Fields 162, 155–198 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bogdan, K., Grzywny, T., Ryznar, M.: Dirichlet heat kernel for unimodal Lévy processes. Stoch. Process. Appl. 124(11), 3612–3650 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chen, Z.-Q., Kim, P., Kumagai, T.: On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces. Acta Math. Sin. Engl. Ser. 25, 1067–1086 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Chen, Z.-Q., Kim, P., Kumagai, T.: Global heat kernel estimates for symmetric jump processes. Trans. Am. Math. Soc. 363, 5021–5055 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chen, Z.-Q., Kim, P., Kumagai, T.: Heat kernel estimates for reflected diffusions with jumps on metric measure spaces (in preparation) Google Scholar
  10. 10.
    Chen, Z.-Q., Kim, P., Song, R.: Heat kernel estimates for Dirichlet fractional Laplacian. J. Eur. Math. Soc. 12, 1307–1329 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, Z.-Q., Kim, P., Song, R.: Heat kernel estimate for \(\Delta+\Delta^{\alpha/2}\) in \(C^{1,1}\) open sets. J. Lond. Math. Soc. 84, 58–80 (2011) MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chen, Z.-Q., Kim, P., Song, R.: Sharp heat kernel estimates for relativistic stable processes in open sets. Ann. Probab. 40, 213–244 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Chen, Z.-Q., Kim, P., Song, R.: Global heat kernel estimates for relativistic stable processes in half-space-like open sets. Potential Anal. 36, 235–261 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Chen, Z.-Q., Kim, P., Song, R.: Global heat kernel estimates for relativistic stable processes in exterior open sets. J. Funct. Anal. 263, 448–475 (2012) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chen, Z.-Q., Kim, P., Song, R.: Dirichlet heat kernel estimates for rotationally symmetric Lévy processes. Proc. Lond. Math. Soc. 109, 90–120 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chen, Z.-Q., Kim, P., Song, R.: Dirichlet heat kernel estimates for subordinate Brownian motions with Gaussian components. J. Reine Angew. Math. 711, 111–138 (2016) MathSciNetzbMATHGoogle Scholar
  17. 17.
    Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for stable-like processes on \(d\)-sets. Stoch. Process. Appl. 108, 27–62 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140, 277–317 (2008) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Chen, Z.-Q., Kumagai, T.: A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps. Rev. Mat. Iberoam. 26, 551–589 (2010) MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Chen, Z.-Q., Tokle, J.: Global heat kernel estimates for fractional Laplacians in unbounded open sets. Probab. Theory Relat. Fields 149, 373–395 (2011) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Davies, E.B.: The equivalence of certain heat kernel and Green function bounds. J. Funct. Anal. 71, 88–103 (1987) MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Grzywny, T.: On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes. Potential Anal. 41, 1–29 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kim, P., Kim, K.-Y.: Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in \(C^{1,\eta}\) open sets. Stoch. Process. Appl. 124, 3055–3083 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Kim, P., Song, R., Vondraček, Z.: Uniform boundary Harnack principle for rotationally symmetric Lévy processes in general open sets. Sci. China Math. 55, 2193–2416 (2012) CrossRefzbMATHGoogle Scholar
  25. 25.
    Knopova, V., Schilling, R.: A note on the existence of transition probability densities of Lévy processes. Forum Math. 25, 125–149 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Kwaśnicki, M., Małecki, J., Ryznar, M.: Suprema of Lévy processes. Ann. Probab. 41, 2047–2065 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pruitt, W.E.: The growth of random walks and Lévy processes. Ann. Probab. 9, 948–956 (1981) MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Zhang, Q.S.: The boundary behavior of heat kernels of Dirichlet Laplacians. J. Differ. Equ. 182, 416–430 (2002) MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of WashingtonSeattleUSA
  2. 2.Department of Mathematical Sciences and Research Institute of MathematicsSeoul National UniversitySeoulRepublic of Korea

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