The Annals of Probability

Heat kernel estimates for the fractional Laplacian with Dirichlet conditions

Krzysztof Bogdan, Tomasz Grzywny, and Michał Ryznar

Full-text: Open access

Abstract

We give sharp estimates for the heat kernel of the fractional Laplacian with Dirichlet condition for a general class of domains including Lipschitz domains.

Article information

Source
Ann. Probab., Volume 38, Number 5 (2010), 1901-1923.

Dates
First available in Project Euclid: 17 August 2010

Permanent link to this document
https://projecteuclid.org/euclid.aop/1282053775

Digital Object Identifier
doi:10.1214/10-AOP532

Mathematical Reviews number (MathSciNet)
MR2722789

Zentralblatt MATH identifier
1204.60074

Subjects
Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J50: Boundary theory
Secondary: 60J75: Jump processes 31B25: Boundary behavior

Keywords
Fractional Laplacian Dirichlet problem heat kernel estimate Lipschitz domain boundary Harnack principle

Citation

Bogdan, Krzysztof; Grzywny, Tomasz; Ryznar, Michał. Heat kernel estimates for the fractional Laplacian with Dirichlet conditions. Ann. Probab. 38 (2010), no. 5, 1901--1923. doi:10.1214/10-AOP532. https://projecteuclid.org/euclid.aop/1282053775


Export citation

References

  • [1] Bañuelos, R. and Bogdan, K. (2004). Symmetric stable processes in cones. Potential Anal. 21 263–288.
  • [2] Bañuelos, R. and Kulczycki, T. (2008). Trace estimates for stable processes. Probab. Theory Related Fields 142 313–338.
  • [3] Barlow, M. T., Grigor’yan, A. and Kumagai, T. (2009). Heat kernel upper bounds for jump processes and the first exit time. J. Reine Angew. Math. 626 135–157.
  • [4] Blumenthal, R. M. and Getoor, R. K. (1960). Some theorems on stable processes. Trans. Amer. Math. Soc. 95 263–273.
  • [5] Blumenthal, R. M., Getoor, R. K. and Ray, D. B. (1961). On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99 540–554.
  • [6] Bogdan, K. (2000). Sharp estimates for the Green function in Lipschitz domains. J. Math. Anal. Appl. 243 326–337.
  • [7] Bogdan, K. and Byczkowski, T. (1999). Potential theory for the α-stable Schrödinger operator on bounded Lipschitz domains. Studia Math. 133 53–92.
  • [8] Bogdan, K. and Byczkowski, T. (2000). Potential theory of Schrödinger operator based on fractional Laplacian. Probab. Math. Statist. 20 293–335.
  • [9] Bogdan, K., Byczkowski, T., Kulczycki, T., Ryznar, M., Song, R. and Vondraček, Z. (2009). Potential Analysis of Stable Processes and Its Extensions (P. Graczyk and A. Stos, eds.). Lecture Notes in Math. 1980. Springer, Berlin.
  • [10] Bogdan, K. and Grzywny, T. (2010). Heat kernel of fractional Laplacian in cones. Colloq. Math. 118 365–377.
  • [11] Bogdan, K., Grzywny, T. and Ryznar, M. (2009). Heat kernel estimates for the fractional Laplacian. Preprint. Available at http://arxiv.org/abs/0905.2626v1.
  • [12] Bogdan, K., Hansen, W. and Jakubowski, T. (2008). Time-dependent Schrödinger perturbations of transition densities. Studia Math. 189 235–254.
  • [13] Bogdan, K. and Jakubowski, T. (2007). Estimates of heat kernel of fractional Laplacian perturbed by gradient operators. Comm. Math. Phys. 271 179–198.
  • [14] Bogdan, K., Kulczycki, T. and Kwaśnicki, M. (2008). Estimates and structure of α-harmonic functions. Probab. Theory Related Fields 140 345–381.
  • [15] Bogdan, K., Kulczycki, T. and Nowak, A. (2002). Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes. Illinois J. Math. 46 541–556.
  • [16] Bogdan, K., Stós, A. and Sztonyk, P. (2003). Harnack inequality for stable processes on d-sets. Studia Math. 158 163–198.
  • [17] Bogdan, K. and Sztonyk, P. (2007). Estimates of the potential kernel and Harnack’s inequality for the anisotropic fractional Laplacian. Studia Math. 181 101–123.
  • [18] Bogdan, K. and Żak, T. (2006). On Kelvin transformation. J. Theoret. Probab. 19 89–120.
  • [19] Chen, Z. Q., Kim, P. and Song, R. (2010). Heat kernel estimates for Dirichlet fractional Laplacian. J. European Math. Soc. To appear.
  • [20] Chen, Z.-Q. and Kumagai, T. (2008). Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields 140 277–317.
  • [21] Chen, Z.-Q. and Song, R. (1998). Estimates on Green functions and Poisson kernels for symmetric stable processes. Math. Ann. 312 465–501.
  • [22] Chen, Z.-Q. and Tokle, J. (2009). Global heat kernel estimates for fractional Laplacians in unbounded open sets. Probab. Theory Related Fields. DOI: 10.1007/s00440-009-0256-0. To appear.
  • [23] Grigor’yan, A. and Hu, J. (2008). Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces. Invent. Math. 174 81–126.
  • [24] Grzywny, T. and Ryznar, M. (2007). Estimates of Green functions for some perturbations of fractional Laplacian. Illinois J. Math. 51 1409–1438.
  • [25] Grzywny, T. and Ryznar, M. (2008). Two-sided optimal bounds for Green functions of half-spaces for relativistic α-stable process. Potential Anal. 28 201–239.
  • [26] Hansen, W. (2006). Global comparison of perturbed Green functions. Math. Ann. 334 643–678.
  • [27] Ikeda, N. and Watanabe, S. (1962). On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes. J. Math. Kyoto Univ. 2 79–95.
  • [28] Jakubowski, T. (2002). The estimates for the Green function in Lipschitz domains for the symmetric stable processes. Probab. Math. Statist. 22 419–441.
  • [29] Kulczycki, T. (1997). Properties of Green function of symmetric stable processes. Probab. Math. Statist. 17 339–364.
  • [30] Kulczycki, T. (1998). Intrinsic ultracontractivity for symmetric stable processes. Bull. Polish Acad. Sci. Math. 46 325–334.
  • [31] Kulczycki, T., Kwaśnicki, M., Małecki, J. and Stós, A. (2009). Spectral properties of the Cauchy process on half-line and interval. Proc. London Math. Soc. DOI: 10.1112/plms/pdq010. To appear.
  • [32] Kulczycki, T. and Siudeja, B. (2006). Intrinsic ultracontractivity of the Feynman–Kac semigroup for relativistic stable processes. Trans. Amer. Math. Soc. 358 5025–5057 (electronic).
  • [33] Kwaśnicki, M. (2009). Intrinsic ultracontractivity for stable semigroups on unbounded open sets. Potential Anal. 31 57–77.
  • [34] Luks, T. (2009). Harmonic Hardy spaces on smooth domains. Preprint. Available at http://arXiv.org/abs/0909.3370v1.
  • [35] Michalik, K. (2006). Sharp estimates of the Green function, the Poisson kernel and the Martin kernel of cones for symmetric stable processes. Hiroshima Math. J. 36 1–21.
  • [36] Rao, M., Song, R. and Vondraček, Z. (2006). Green function estimates and Harnack inequality for subordinate Brownian motions. Potential Anal. 25 1–27.
  • [37] Riesz, M. (1938). Intégrales de Riemann–Liouville et potentiels. Acta Sci. Math. Szeged. 9 1–42.
  • [38] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
  • [39] Siudeja, B. (2006). Symmetric stable processes on unbounded domains. Potential Anal. 25 371–386.
  • [40] Song, R. and Wu, J.-M. (1999). Boundary Harnack principle for symmetric stable processes. J. Funct. Anal. 168 403–427.
  • [41] Varopoulos, N. T. (2003). Gaussian estimates in Lipschitz domains. Canad. J. Math. 55 401–431.
  • [42] Zhang, Q. S. (2002). The boundary behavior of heat kernels of Dirichlet Laplacians. J. Differential Equations 182 416–430.
  • [43] Zhao, Z. X. (1986). Green function for Schrödinger operator and conditioned Feynman–Kac gauge. J. Math. Anal. Appl. 116 309–334.