Document ESAIM: P&S, August 2007, Vol. 11, pp. 327-343
DOI: 10.1051/ps:2007022
Small ball probabilities for stable convolutions
Frank Aurzada1 and Thomas Simon21 Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany; aurzada@math.tu-berlin.de
2 Equipe d'analyse et probabilités, Université d'Evry-Val d'Essonne, boulevard François Mitterrand, 91025 Evry Cedex, France; tsimon@univ-evry.fr
(Received April 4, 2006. Revised July 10, 2006. Published online 17 August 2007.)
Abstract
We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function
![$f : \; ]0, +\infty[ \;\to \mathbb{R} $](/articles/ps/abs/2007/01/ps0624/img1.gif){kind=small}
with a real 
Lévy process. We show that the small ball exponent is uniquely determined by the norm and by the behaviour of f at zero, which extends the results of
Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 725-752
where this was proved for f being a power function (Riemann-Liouville processes). In the Gaussian case, the same generality as Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005) 725-752 is obtained with respect to the norms, thanks to a weak decorrelation inequality due to Li, Elec. Comm. Probab. 4 (1999) 111-118. In the more difficult non-Gaussian case, we use a different method relying on comparison of entropy numbers and restrict ourselves to Hölder and Lp-norms.
Mathematics Subject Classification. 60F99, 60G15, 60G20, 60G52
Key words: Entropy numbers, fractional Ornstein-Uhlenbeck processes, Riemann-Liouville processes, small ball probabilities, stochastic convolutions, wavelets.
© EDP Sciences, SMAI 2007