Almost everywhere convergence of convolution powers on compact abelian groups

Conze, Jean-Pierre; Lin, Michael

doi:10.1214/11-AIHP468

Conze, Jean-Pierre ; Lin, Michael

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 550-568.

Résumé
Abstract

Il est connu qu’une mesure de probabilité $μ$ sur le cercle $𝕋$ satisfait $∥ μ^{n} {* f - \int f d m ∥}_{p} \to 0$ pour toute fonction $f \in L_{p}$ et pour tout $p \in [1, \infty)$ (ou pour un $p \in [1, \infty)$ ), si et seulement si $μ$ est strictement apériodique (i.e. $| \hat{μ} (n) | < 1$ pour tout $n$ non nul dans $ℤ$ ). Nous étudions ici la convergence presque partout de $μ^{n} * f$ pour $f \in L_{p}$ , $p > 1$ . Nous montrons une condition nécessaire et suffisante portant sur les coefficients de Fourier-Stieltjes de $μ$ pour la propriété de “balayage fort” (existence d’un borélien $B$ tel que $lim sup μ^{n} * 1_{B} = 1$ p.p. et $lim inf μ^{n} * 1_{B} = 0$ p.p.). Les résultats sont étendus aux groupes abéliens compacts généraux $G$ de mesure de Haar $m$ . Comme corollaire nous obtenons la dichotomie suivante : pour $μ$ strictement apériodique, soit $μ^{n} * f \to \int f d m$ p.p. pour tout $p > 1$ et toute fonction $f \in L_{p} (G, m)$ , soit $μ$ vérifie la propriété de balayage fort.

It is well-known that a probability measure $μ$ on the circle $𝕋$ satisfies $∥ μ^{n} {* f - \int f d m ∥}_{p} \to 0$ for every $f \in L_{p}$ , every (some) $p \in [1, \infty)$ , if and only if $| \hat{μ} (n) | < 1$ for every non-zero $n \in ℤ$ ( $μ$ is strictly aperiodic). In this paper we study the a.e. convergence of $μ^{n} * f$ for every $f \in L_{p}$ whenever $p > 1$ . We prove a necessary and sufficient condition, in terms of the Fourier-Stieltjes coefficients of $μ$ , for the strong sweeping out property (existence of a Borel set $B$ with $lim sup μ^{n} * 1_{B} = 1$ a.e. and $lim inf μ^{n} * 1_{B} = 0$ a.e.). The results are extended to general compact Abelian groups $G$ with Haar measure $m$ , and as a corollary we obtain the dichotomy: for $μ$ strictly aperiodic, either $μ^{n} * f \to \int f d m$ a.e. for every $p > 1$ and every $f \in L_{p} (G, m)$ , or $μ$ has the strong sweeping out property.

MR Zbl

DOI : 10.1214/11-AIHP468

Classification : 37A30, 28D05, 47A35, 60G50, 42A38
Mots-clés : convolution powers, almost everywhere convergence, sweeping out, strictly aperiodic probabilities

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     title = {Almost everywhere convergence of convolution powers on compact abelian groups},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {550--568},
     publisher = {Gauthier-Villars},
     volume = {49},
     number = {2},
     year = {2013},
     doi = {10.1214/11-AIHP468},
     mrnumber = {3088381},
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     language = {en},
     url = {https://www.numdam.org/articles/10.1214/11-AIHP468/}
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Conze, Jean-Pierre; Lin, Michael. Almost everywhere convergence of convolution powers on compact abelian groups. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 2, pp. 550-568. doi : 10.1214/11-AIHP468. https://www.numdam.org/articles/10.1214/11-AIHP468/

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