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**A spectral Radius Formula for the
Fourier Transform on a Compact Group
with Applications to Random Walks**

Dimitris Gatzouras

Agricultural Univ. of Athens

Τρίτη 22 Νοεμβρίου 2005, ώρα 12-1, Αίθουσα Ζ 301

Dimitris Gatzouras

Agricultural Univ. of Athens

Τρίτη 22 Νοεμβρίου 2005, ώρα 12-1, Αίθουσα Ζ 301

If is a compact group, and a regular Borel probability measure on , it is known that the random walk on whose steps are distributed according to the measure converges in distribution to the Haar measure of the group (uniform distribution) iff is aperiodic, i.e., not concentrated on a proper closed subgroup of or on a coset of a proper closed normal subgroup. This was proved by P. Levy for the group (the unit circle) and then established by others for more general compact groups. Notice that the distribution of the position of the random walk at time is the -fold convolution , and ``convergence in distribution" in the above statement means that (Haar measure) in the weak topology. It is also known that this convergence also holds in the total variation norm, i.e., , if in addition not all powers of are singular with respect to Haar measure . We shall derive these results by means of Fourier transforms. To prove the convergence in the total variation norm, we shall first obtain spectral radii formulae for the Fourier transforms of functions and measures, analoguous to the Beurling-Gelfand spectral radius formula for the Fourier transform in Abelian groups. As another application of the spectral radius formula for functions, one obtains the uniform convergence (to ) of the densities of the measures when has a density with respect to Haar measure which belongs to for some , a result first obtained by Shlosman by a different method.

Joint work with M. Annousis.

`http://fourier.math.uoc.gr/~ seminar`

Analysis Seminar 2005-11-15