Small Deviations of Riemann-Liouville Processes In lq Spaces With Respect To Fractal Measures |
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Authors: | Lifshits, Mikhail A. Linde, Werner Shi, Zhan |
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Affiliation: | St Petersburg State University Department of Mathematics, and Mechanics 198504 Stary Peterhof, Bibliotechnaya pl., 2, Russia lifts{at}mail.rcom.ru FSU Jena, Institut für Stochastik ErnstAbbePlatz 2, 07743 Jena, Germany lindew{at}minet.uni-jena.de Laboratoire de Probabilités et, Modèles Aléatoires, Université Paris VI 4 place Jussieu, F-75252 Paris Cedex 05, France zhan{at}proba.jussieu.fr |
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Abstract: | We investigate RiemannLiouville processes RH, with H> 0, and fractional Brownian motions BH, for 0 < H <1, and study their small deviation properties in the spacesLq([0, 1], µ). Of special interest here are thin (fractal)measures µ, that is, those that are singular with respectto the Lebesgue measure. We describe the behavior of small deviationprobabilities by numerical quantities of µ, called mixedentropy numbers, characterizing size and regularity of the underlyingmeasure. For the particularly interesting case of self-similarmeasures, the asymptotic behavior of the mixed entropy is evaluatedexplicitly. We also provide two-sided estimates for this quantityin the case of random measures generated by subordinators. While the upper asymptotic bound for the small deviation probabilityis proved by purely probabilistic methods, the lower bound isverified by analytic tools concerning entropy and Kolmogorovnumbers of RiemannLiouville operators. 2000 MathematicsSubject Classification 60G15 (primary), 47B06, 47G10, 28A80(secondary). |
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