Large time behaviour of heat kernels on non-compact manifolds: fast and slow decays

Thierry Coulhon

Journées équations aux dérivées partielles (1998)

  • page 1-12
  • ISSN: 0752-0360

Abstract

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In this talk we shall present some joint work with A. Grigory’an. Upper and lower estimates on the rate of decay of the heat kernel on a complete non-compact riemannian manifold have recently been obtained in terms of the geometry at infinity of the manifold, more precisely in terms of a kind of L 2 isoperimetric profile. The main point is to connect the decay of the L 1 - L ∞ norm of the heat semigroup with some adapted Nash or Faber-Krahn inequalities, which is done by functional analytic methods. We shall give an outline of these results and show how they can give some answers to the following question: given the volume growth of a manifold, e.g. polynomial or exponential, how fast and how slow can the heat kernel decay be?

How to cite

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Coulhon, Thierry. "Large time behaviour of heat kernels on non-compact manifolds: fast and slow decays." Journées équations aux dérivées partielles (1998): 1-12. <http://eudml.org/doc/93359>.

@article{Coulhon1998,
abstract = {In this talk we shall present some joint work with A. Grigory’an. Upper and lower estimates on the rate of decay of the heat kernel on a complete non-compact riemannian manifold have recently been obtained in terms of the geometry at infinity of the manifold, more precisely in terms of a kind of $L^2$ isoperimetric profile. The main point is to connect the decay of the $L^1-L^\infty $ norm of the heat semigroup with some adapted Nash or Faber-Krahn inequalities, which is done by functional analytic methods. We shall give an outline of these results and show how they can give some answers to the following question: given the volume growth of a manifold, e.g. polynomial or exponential, how fast and how slow can the heat kernel decay be?},
author = {Coulhon, Thierry},
journal = {Journées équations aux dérivées partielles},
keywords = {upper estimates; lower estimates; rate of decay; complete noncompact Riemannian manifold; Poincaré type inequalities},
language = {eng},
pages = {1-12},
publisher = {Université de Nantes},
title = {Large time behaviour of heat kernels on non-compact manifolds: fast and slow decays},
url = {http://eudml.org/doc/93359},
year = {1998},
}

TY - JOUR
AU - Coulhon, Thierry
TI - Large time behaviour of heat kernels on non-compact manifolds: fast and slow decays
JO - Journées équations aux dérivées partielles
PY - 1998
PB - Université de Nantes
SP - 1
EP - 12
AB - In this talk we shall present some joint work with A. Grigory’an. Upper and lower estimates on the rate of decay of the heat kernel on a complete non-compact riemannian manifold have recently been obtained in terms of the geometry at infinity of the manifold, more precisely in terms of a kind of $L^2$ isoperimetric profile. The main point is to connect the decay of the $L^1-L^\infty $ norm of the heat semigroup with some adapted Nash or Faber-Krahn inequalities, which is done by functional analytic methods. We shall give an outline of these results and show how they can give some answers to the following question: given the volume growth of a manifold, e.g. polynomial or exponential, how fast and how slow can the heat kernel decay be?
LA - eng
KW - upper estimates; lower estimates; rate of decay; complete noncompact Riemannian manifold; Poincaré type inequalities
UR - http://eudml.org/doc/93359
ER -

References

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  1. [1] Bakry D., Coulhon T., Ledoux M., Saloff-Coste L., Sobolev inequalities in disguise, Indiana Univ. Math. J., 44, 4, 1033-1074, 1995. Zbl0857.26006MR97c:46039
  2. [2] Carron G., Inégalités isopérimétriques sur les variétés riemanniennes. Thesis, University of Grenoble, 1994. 
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  10. [10] Coulhon T., Grigor'yan A., Manifolds with big heat kernels, preprint. 
  11. [11] Coulhon T., Ledoux M., Isopérimétrie, décroissance du noyau de la chaleur et transformations de Riesz : un contre-exemple, Arkiv för Mat., 32, 63-77, 1994. Zbl0826.53035MR95e:58170
  12. [12] Coulhon T., Saloff-Coste L., Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamer., 9, 2, 293-314, 1993. Zbl0782.53066MR94g:58263
  13. [13] Coulhon T., Saloff-Coste L., Variétés riemanniennes isométriques à l'infini, Rev. Mat. Iberoamer., 11, 3, 687-726, 1995. Zbl0845.58054MR96m:53035
  14. [14] Coulhon T., Saloff-Coste L., Harnack inequality and hyperbolicity for the p- Laplacian with applications to Picard type theorems, preprint. Zbl1005.58013
  15. [15] Grigor'yan A., The heat equation on non-compact Riemannian manifolds, in Russian : Matem. Sbornik, 182, 1, 55-87, 1991 ; English translation : Math. USSR Sb., 72, 1, 47-77, 1992. Zbl0776.58035MR92h:58189
  16. [16] Grigor'yan A., Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana, 10, 2, 395-452, 1994. Zbl0810.58040MR96b:58107
  17. [17] Grigor'yan A., Heat kernel on a non-compact Riemannian manifold, in 1993 Summer research institute on stochastic analysis, ed. M. Pinsky et alia, Proceedings of Symposia in Pure Math., 57, 239-263, 1994. Zbl0829.58041MR96f:58155
  18. [18] Pittet C., Saloff-Coste L., Amenable groups, isoperimetric profiles, and random walks, in Proceedings of the 1996 Canberra Geometric group theory conference, 1997. Zbl0934.43001

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