Journal d’Analyse Mathématique

, Volume 58, Issue 1, pp 99–119 | Cite as

Heat kernel bounds, conservation of probability and the feller property

  • E. B. Davies


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  1. [1]
    S. Albeverio and M. Rockner,Classical Dirichlet forms on topological vector spaces—closability and a Cameron-Martin formula, J. Funct. Anal., to appear.Google Scholar
  2. [2]
    A. Ancona, University of Warwick Lectures, December, 1989.Google Scholar
  3. [3]
    R. Azencott,Behaviour of diffusion semigroups at infinity, Bull. Soc. Math. Fr.102 (1974), 193–240.MATHMathSciNetGoogle Scholar
  4. [4]
    I. Chavel,Eigenvalues in Riemannian Geometry, Academic Press, New York, 1984.MATHGoogle Scholar
  5. [5]
    J. Cheeger, M. Gromov and M. Taylor,Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differ. Geom.17 (1982), 15–53.MATHMathSciNetGoogle Scholar
  6. [6]
    F. Chiarenza and R. Serapioni,A remark on a Harnack inequality for, degenerate parabolic equations, Rend. Sem. Mat. Univ. Padova73 (1985), 179–190.MathSciNetMATHGoogle Scholar
  7. [7]
    E. B. Davies,Gaussian upper bounds for the heat kernels of some second order operators on Riemannian manifolds, J. Funct. Anal.80 (1988), 16–32.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    E. B. Davies,Heat Kernels and Spectral Theory, Cambridge University Press, 1989.Google Scholar
  9. [9]
    E. B. Davies and M. M. H. Pang,Sharp heat kernel bounds for some Laplace operators, Quart. J. Math. Oxford (2)40 (1989), 281–290.MATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    E. B. Fabes and D. W. Stroock,A new proof of Moser's parabolic Harnack inequality via the old ideas of Nash, Arch. Ration. Mech. Anal.96 (1986), 327–338.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    M. Fukushima,Dirichlet Forms and Markov Processes, North-Holland, Amsterdam, 1980.MATHGoogle Scholar
  12. [12]
    M. P. Gaffney,The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math.12 (1959), 1–11.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    A. A. Grigor'yan,On stochastically complete Riemannian manifolds, Soviet Math. Doklady34 (1987), 310–313.MATHGoogle Scholar
  14. [14]
    R. Z. Hasminkii,Ergodic properties of recurrent diffusion processes and stabilisation of the solution of the Cauchy problem for parabolic equations, Theor. Probab. Appl.5 (1960), 179–196.CrossRefGoogle Scholar
  15. [15]
    K. Ichihara,Curvature, geodesics and the Brownian motion on a Riemannian manifold II, explosion properties, Nagoya Math. J.87 (1982), 115–125.MATHMathSciNetGoogle Scholar
  16. [16]
    K. Ichihara,Explosion problems for symmetric, diffusion processes, Proc. Japan Acad.60 (1984), 243–245.MATHMathSciNetCrossRefGoogle Scholar
  17. [17]
    K. Ichihara,Explosion problems for symmetric diffusion processes, Trans. Am. Math. Soc.298 (1986), 515–536.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    P. Li and L. Karp,The heat equation on complete Riemannian manifolds, preprint, 1982.Google Scholar
  19. [19]
    P. Li and S. T. Yau,On the parabolic kernel of the Schrödinger operator, Acta Math.156 (1986), 153–201.CrossRefMathSciNetGoogle Scholar
  20. [20]
    M. Lianantonakis, preprint, in preparation.Google Scholar
  21. [21]
    T. Lyons and W. Zheng,Crossing estimate for canonical process on a Dirichlet space and a tightness result, preprint 1988: Asterique, to appear.Google Scholar
  22. [22]
    J. Moser,On a pointwise estimate for parabolic differential equations, Commun. Pure Appl. Math.24 (1971), 727–740.MATHCrossRefGoogle Scholar
  23. [23]
    Y. Oshima,On conservativeness and recurrence criteria of the Markov processes, preprint, 1989.Google Scholar
  24. [24]
    M. M. H. Pang,L 1 properties of two classes of singular second order elliptic operators, J. London Math. Soc. (2)38 (1988), 525–543.MATHMathSciNetGoogle Scholar
  25. [25]
    M. M. H. Pang, L1 and L2 spectral properties of a class of singular second order elliptic operators with measurable coefficients on \({\mathcal{R}}^N \) preprint, 1990.Google Scholar
  26. [26]
    M. Takeda,On a martingale method for symmetric diffusion processes and its applications, Osaka J. Math., to appear.Google Scholar
  27. [27]
    M. Takeda,The conservation property for the Brownian motion on Riemannian manifolds, preprint, 1989; Bull. London Math. Soc., to appear.Google Scholar
  28. [28]
    N. Teleman,The index of signature operators on Lipschitz manifolds, Publ. Math. Inst. Hautes Etudes Sci.58 (1983), 39–78.MathSciNetCrossRefGoogle Scholar
  29. [29]
    N. Th. Varopoulos,Potential theory and diffusion on Riemannian manifolds, inProc. Conf. on Harmonic Analysis in Honor of A. Zygmund, Chicago (W. Beckneret al., eds.), Wadsworth, 1983, pp. 821–837.Google Scholar
  30. [30]
    S. T. Yau,On the heat kernel of a complete Riemannian manifold, J. Math. Pures Appl.57 (1978), 191–201.MATHMathSciNetGoogle Scholar

Copyright information

©  0246 V 2 1992

Authors and Affiliations

  • E. B. Davies
    • 1
  1. 1.Department of MathematicsKing's CollegeLondonEngland

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