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Σεμινάριο Ανάλυσης
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

If $G$ is a compact group, and $\mu$ a regular Borel probability measure on $G$, it is known that the random walk on $G$ whose steps are distributed according to the measure $\mu$ converges in distribution to the Haar measure of the group (uniform distribution) iff $\mu$ is aperiodic, i.e., not concentrated on a proper closed subgroup of $G$ or on a coset of a proper closed normal subgroup. This was proved by P. Levy for the group $G=S^1$ (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 $n$ is the $n$-fold convolution $\mu^n:=\mu\ast\cdots\ast\mu$, and ``convergence in distribution" in the above statement means that $\mu^n\to\lambda$ (Haar measure) in the weak$^\ast$ topology. It is also known that this convergence also holds in the total variation norm, i.e., $\Vert{\mu^n-\lambda}\Vert\to 0$, if in addition not all powers $\mu^n$ of $\mu$ are singular with respect to Haar measure $\lambda$. 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 $L^1$ 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 $L^1$ functions, one obtains the uniform convergence (to $1$) of the densities of the measures $\mu^n$ when $\mu$ has a density $f$ with respect to Haar measure which belongs to $L^{1+\epsilon}(G)$ for some $\epsilon>0$, a result first obtained by Shlosman by a different method.

Joint work with M. Annousis.

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