The Annals of Probability

Sharp explicit lower bounds of heat kernels

Feng-Yu Wang

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Abstract

By using logarithmic transformations, an explicit lower bound estimate of heat kernels is obtained for diffusion processes on Riemannian manifolds. This estimate is sharp for both short and long times, especially for heat kernels on a compact manifold, and is extended to manifolds with unbounded curvature.

Article information

Source
Ann. Probab., Volume 25, Number 4 (1997), 1995-2006.

Dates
First available in Project Euclid: 7 June 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1023481118

Digital Object Identifier
doi:10.1214/aop/1023481118

Mathematical Reviews number (MathSciNet)
MR1487443

Zentralblatt MATH identifier
0918.58070

Subjects
Primary: 58G11 60H10

Keywords
Heat kernel logarithmic transformation diffusion process

Citation

Wang, Feng-Yu. Sharp explicit lower bounds of heat kernels. Ann. Probab. 25 (1997), no. 4, 1995--2006. doi:10.1214/aop/1023481118. https://projecteuclid.org/euclid.aop/1023481118


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References

  • [1] Bakry, D. and Qian,(1996). Gradient estimates and Harnack inequality on a complete manifold. Preprint.
  • [2] Berg, M., Gauduchon, P. and Mazet, E. (1971). Le spectre d'une vari´et´e Riemannienne. Lecture Notes in Math. 194. Springer, Berlin.
  • [3] Benjamini, I., Chavel, I. and Feldman, E. A. (1996). Heat kernel lower bounds on Riemannian manifolds using the old ideas of Nash. Proc. London Math. Soc. 72 215-240.
  • [4] Benjamini, I., Chavel, I. and Feldman, E. A. (1995). Heat kernel lower bounds on Riemannian manifolds. Proc. Sympos. Pure Math. 57 213-225.
  • [5] Chavel, I. (1984). Eigenvalues in Riemannian geometry. Academic Press, New York.
  • [6] Cheeger, J. and Yau, S. T. (1981). A lower bound for heat kernel. Comm. Pure Applied Math. 34 465-480.
  • [7] Cranston, M. (1991). Gradient estimates on manifolds using coupling. J. Funct. Anal. 99 110-124.
  • [8] Davies, E. B. (1989). Heat Kernels and Spectral Theory. Cambridge Univ. Press.
  • [9] Davies, E. B. and Mandouvalos, N. (1988). Heat kernel bounds on hyperbolic space and Kleinian groups Proc. London Math. Soc. 57 182-208.
  • [10] Fleming, W. H. (1982). Logarithmic transformations and stochastic control. Lecture Notes in Control and Inform. Sci. 42 131-141.
  • [11] Grigor'yan, A. A. (1994). Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoamericana 10 395-452.
  • [12] Grigor'yan, A. A. (1994). Heat kernel on a manifold with a local Harnack inequality. Comm. Anal. Geom. 2 111-138.
  • [13] Kendall, W. S. (1987). The radial part of Brownian motion on a manifold: a semimartingale property. Ann. Probab. 15 1491-1500.
  • [14] Li, P. (1986). Large time behaviour of the heat equation on complete manifolds with nonnegative curvature. Ann. Math. 124 1-21.
  • [15] Li, P. and Yau, S. T. (1986). On the parabolic heat kernel of the Schr¨odinger operator. Acta Math. 156 153-201.
  • [16] Qian,(1995). Gradient estimates and heat kernel estimates. Proc. Roy. Soc. Edinburgh Sect. A 125 975-990.
  • [17] Setti, A. G. (1992). Gaussian estimates for the heat kernel of the weighted Laplacian and fractal measures. Canad. J. Math. 44 1061-1078.
  • [18] Sheu, S. J. (1991). Some estimates of the transition density of a nondegenerate diffusion Markov process. Ann. Probab. 19 538-561.
  • [19] Wang, F. Y. (1994). Application of coupling method to the Neumann eigenvalue problem. Probab. Theory Related Fields 98 299-306.
  • [20] Yau, S. T. (1995). Harnack inequality for non-self-adjoint evolution equations. Math. Res. Lett. 2 387-399.