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Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups

  Published:2006-08-01
 Printed: Aug 2006
  • A. Bendikov
  • L. Saloff-Coste
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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 infinite-dimensional 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 infinite-dimensional spaces [See also 58C35]
60J45 - Probabilistic potential theory [See also 31Cxx, 31D05]
60J60 - Diffusion processes [See also 58J65]
 

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