 
Summary: Probability Measures on Compact Groups
which have SquareIntegrable Densities
David Applebaum
Abstract
We apply PeterWeyl theory to obtain necessary and sufficient
conditions for a probability measure on a compact group to have a
squareintegrable density. Applications are given to measures on the
ddimensional torus.
MSC 2000: 60B15, 60E07, 43A05, 43A30
1 Introduction
Given a probability measure on a Riemannian manifold it is highly beneficial
if we can establish that it has a density with respect to the Riemannian
volume measure. This is vital for statistical inference but a density may
also contain important information about the topology and geometry of the
manifold, e.g. the heat kernel is the density of Brownian motion on a manifold
(see e.g. [6], [18].) Much work by stochastic analysts has been focussed
on finding densities for solutions of stochastic differential equations driven
by Brownian motion. A key result here is H¨ormander's theorem [10] which
gives a sufficient condition (called "hypoellipticity") for existence of a smooth
density involving the Lie algebra generated by the driving vector fields. The
