IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v118y2008i3p368-388.html
   My bibliography  Save this article

Hypoelliptic heat kernel inequalities on Lie groups

Author

Listed:
  • Melcher, Tai

Abstract

This paper discusses the existence of gradient estimates for the heat kernel of a second order hypoelliptic operator on a manifold. For elliptic operators, it is now standard that such estimates (satisfying certain conditions on coefficients) are equivalent to a lower bound on the Ricci tensor of the Riemannian metric. For hypoelliptic operators, the associated "Ricci curvature" takes on the value -[infinity] at points of degeneracy of the semi-Riemannian metric. For this reason, the standard proofs for the elliptic theory fail in the hypoelliptic setting. This paper presents recent results for hypoelliptic operators. Malliavin calculus methods transfer the problem to one of determining certain infinite dimensional estimates. Here, the underlying manifold is a Lie group, and the hypoelliptic operators are given by the sum of squares of left invariant vector fields. In particular, "Lp-type" gradient estimates hold for p[set membership, variant](1,[infinity]), and the p=2 gradient estimate implies a Poincaré estimate in this context.

Suggested Citation

  • Melcher, Tai, 2008. "Hypoelliptic heat kernel inequalities on Lie groups," Stochastic Processes and their Applications, Elsevier, vol. 118(3), pages 368-388, March.
  • Handle: RePEc:eee:spapps:v:118:y:2008:i:3:p:368-388
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(07)00072-5
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:118:y:2008:i:3:p:368-388. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.