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Gaussian bounds for derivatives of central Gaussian semigroups on compact groups


Authors: A. Bendikov and L. Saloff-Coste
Journal: Trans. Amer. Math. Soc. 354 (2002), 1279-1298
MSC (2000): Primary 60B15, 43A77, 60J45, 60J60
DOI: https://doi.org/10.1090/S0002-9947-01-02945-2
Published electronically: November 19, 2001
MathSciNet review: 1873006
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Abstract | References | Similar Articles | Additional Information

Abstract: For symmetric central Gaussian semigroups on compact connected groups, assuming the existence of a continuous density, we show that this density admits space derivatives of all orders in certain directions. Under some additional assumptions, we prove that these derivatives satisfy certain Gaussian bounds.


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  • 1. Bakry D. Transformation de Riesz pour les semigroupes symétriques. Séminaire de Probabilités XIX, Lect. Notes in Math. 1123, Springer, 1985, 130-174. MR 89h:42023
  • 2. Bendikov A. Potential Theory on Infinite Dimensional Abelian Groups. Walter De Gruyter & Co., Berlin-New York, 1995. MR 98i:31007
  • 3. Bendikov A. Symmetric stable semigroups on the infinite dimensional torus. Expo. Math., 13, 1995, 39-79. MR 96h:60123
  • 4. Bendikov A. and Saloff-Coste L. Elliptic diffusions on infinite products. J. reine angew. Math. 493, 1997, 171-220. MR 99k:58173
  • 5. Bendikov A. and Saloff-Coste L. Potential theory on infinite products and groups. Potential Analysis, 11, 325-358, 1999. MR 2001a:31012
  • 6. Bendikov A. and Saloff-Coste L. On and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces. American J. Math. 122, 2000, 1205-1263. CMP 2001:05
  • 7. Bendikov A. and Saloff-Coste L. Gaussian semigroups of measures on locally compact locally connected groups, Preprint 1999.
  • 8. Bendikov A. and Saloff-Coste L. Central Gaussian semigroups of measures with continuous density, J. Funct. Anal. to appear.
  • 9. Bendikov A. and Saloff-Coste L. On the absolute continuity of Gaussian measures on locally compact groups, J. Theoret. Prob. to appear.
  • 10. Bendikov A. and Saloff-Coste L. On the hypoellipticity of sub-Laplacians on infinite dimensional compact groups. Forum Math. to appear.
  • 11. Bendikov A. and Saloff-Coste L. Hypoellipticity for bi-invariant Laplacians on infinite dimensional compact groups. Preprint 2001.
  • 12. Berg Ch. Potential theory on the infinite dimensional torus. Invent. Math. 32, 1976, 49-100. MR 53:5915
  • 13. Born E. Projective Lie Algebra Bases of a Locally Compact Group and Uniform Differentiability. Math. Zeit. 200, 1989, 279-292. MR 90f:22008
  • 14. Born E. An Explicit Lévy-Hincin Formula for Convolution Semigroups on Locally Compact Groups. J. Theor. Prob. 2, 1989, 325-342. MR 91a:60023
  • 15. Bruhat F. Distributions sur un groupe localement compact et application à l'étude des représentations des groupes $p$-adiques. Bull. Soc. Math. France, 89, 1961, 43-75. MR 25:4354
  • 16. Davies E.B. Heat kernels and spectral theory. 1989, Cambridge University Press. MR 90e:35123
  • 17. Davies E.B. Non-Gaussian aspects of heat kernel behaviour. J. London Mat. Soc. 55, 1997, 105-125. MR 97i:58169
  • 18. Fukushima M., Oshima Y. and Takeda, M. Dirichlet forms and symmetric Markov processes, W. De Gruyter, 1994. MR 96f:60126
  • 19. Glushkov V.N. The structure of locally compact groups and Hilbert's Fifth Problem. AMS Translations 15, 1960, 55-94. MR 22:5690
  • 20. Heyer H. Probability Measures on Locally Compact Groups. Ergeb. der Math. und ihrer Grenzgeb. 94, Springer, Berlin-Heildelberg-New York, 1977. MR 58:18648
  • 21. Hofmann K. and Morris S. The structure of compact groups. W. de Gruyter, 1998. MR 99k:22001
  • 22. Hörmander L. Hypoelliptic second order differential equations. Acta Math. 119, 1967, 147-171. MR 36:5526
  • 23. Hunt G. A. Semi-groups of measures on Lie groups. Trans. A.M.S., 81, 1956, 264-293. MR 18:54a
  • 24. Kusuoka S. and Stroock D. Applications of the Malliavin calculus, Part II. J. Fac. Sci., The University of Tokyo, Sec IA, 32, 1985, 1-76. MR 86k:60100b
  • 25. Ledoux M. L'algèbre des gradients itérés d'un générateur Markovien -- Développements de moyennes et entropies. Ann. scient. Éc. Norm. Sup. 4ème série, 28, 1995, 435-460. MR 96f:60130
  • 26. Saloff-Coste L. Analyse sur les groupes de Lie nilpotents. C. R. Acad. Sci. Paris, 302, Série I, 1986, 499-502. MR 88j:22008
  • 27. Siebert E. Absolute continuity, singularity and supports of Gaussian semigroups on a Lie group. Monatshefte für Math. 93, 1982, 239-253. MR 83m:60018
  • 28. Sturm K-T. On the geometry defined by Dirichlet forms. In Seminar on Stochastic Processes, Random Fields and Applications, Ascona, E. Bolthausen et al., Eds. Birkhäuser, Progress in Probability, Vol 36, 1995, 231-242. MR 96j:31009
  • 29. Varopoulos N., Saloff-Coste L. and Coulhon T., Analysis and geometry on groups. Cambridge University Press, 1992. MR 95f:43008

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Additional Information

A. Bendikov
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853-4201
Email: bendikov@math.cornell.edu

L. Saloff-Coste
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853-4201
Email: lsc@math.cornell.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02945-2
Received by editor(s): May 21, 2001
Received by editor(s) in revised form: September 4, 2001
Published electronically: November 19, 2001
Additional Notes: Research partially supported by NSF Grant DMS-9802855
Article copyright: © Copyright 2001 American Mathematical Society

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