Abstract view
Hypoelliptic BiInvariant Laplacians on Infinite Dimensional Compact Groups


Published:20060801
Printed: Aug 2006
A. Bendikov
L. SaloffCoste
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, 43A77, 35H10, 46F25, 60J45, 60J60 show english descriptions
Probability measures on groups or semigroups, Fourier transforms, factorization Analysis on general compact groups Hypoelliptic equations Distributions on infinitedimensional spaces [See also 58C35] Probabilistic potential theory [See also 31Cxx, 31D05] Diffusion processes [See also 58J65]
60B15  Probability measures on groups or semigroups, Fourier transforms, factorization 43A77  Analysis on general compact groups 35H10  Hypoelliptic equations 46F25  Distributions on infinitedimensional spaces [See also 58C35] 60J45  Probabilistic potential theory [See also 31Cxx, 31D05] 60J60  Diffusion processes [See also 58J65]
