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Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups |
Canad. J. Math. 58(2006), 691-725
Published:2006-08-01
Printed: Aug 2006
Printed: Aug 2006
- A. Bendikov
- L. Saloff-Coste
Abstract
On a compact connected group $G$, consider the infinitesimal
generator $-L$ of a central symmetric Gaussian convolution
semigroup $(\mu_t)_{t>0}$. Using appropriate notions of distribution
and smooth function spaces, we prove that $L$ is hypoelliptic if and only if
$(\mu_t)_{t>0} $ is absolutely continuous with respect to Haar measure
and admits a continuous density $x\mapsto \mu_t(x)$, $t>0$, such that
$\lim_{t\ra 0} t\log \mu_t(e)=0$. In particular, this condition holds
if and only if any Borel measure $u$ which is solution of $Lu=0$
in an open set $\Omega$ can be represented by a continuous
function in $\Omega$. Examples are discussed.
| MSC Classifications: |
60B15 - Probability measures on groups or semigroups, Fourier transforms, factorization 43A77 - Analysis on general compact groups 35H10 - Hypoelliptic equations 46F25 - Distributions on infinite-dimensional spaces [See also 58C35] 60J45 - Probabilistic potential theory [See also 31Cxx, 31D05] 60J60 - Diffusion processes [See also 58J65] |