SMALL DEVIATIONS OF RIEMANN–LIOUVILLE PROCESSES IN Lq–SPACES WITH RESPECT TO FRACTAL MEASURES
نویسنده
چکیده
We investigate Riemann–Liouville processes RH ,H > 0, and fractional Brownian motions BH , 0 < H < 1, and study their small deviation properties in the spaces Lq([0, 1], μ). Of special interest are hereby thin (fractal) measures μ, i.e., those which are singular with respect to the Lebesgue measure. We describe the behavior of small deviation probabilities by numerical quantities of μ, called mixed entropy numbers, characterizing size and regularity of the underlying measure. For the particularly interesting case of self–similar measures the asymptotic behavior of the mixed entropy is evaluated explicitly. We also provide two–sided estimates for this quantity in the case of random measures generated by subordinators. While the upper asymptotic bound for the small deviation probability is proved by purely probabilistic methods, the lower bound is verified by analytic tools concerning entropy and Kolmogorov numbers of Riemann–Liouville operators.
منابع مشابه
Kolmogorov numbers of Riemann-Liouville operators over small sets and applications to Gaussian processes
We investigate compactness properties of the Riemann–Liouville operator Rα of fractional integration when regarded as operator from L2[0, 1] into C(K), the space of continuous functions over a compact subset K in [0, 1]. Of special interest are small sets K, i.e. those possessing Lebesgue measure zero (e.g. fractal sets). We prove upper estimates for the Kolmogorov numbers of Rα against certain...
متن کاملNew operational matrix for solving a class of optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative
In this paper, we apply spectral method based on the Bernstein polynomials for solving a class of optimal control problems with Jumarie’s modified Riemann-Liouville fractional derivative. In the first step, we introduce the dual basis and operational matrix of product based on the Bernstein basis. Then, we get the Bernstein operational matrix for the Jumarie’s modified Riemann-Liouville fractio...
متن کاملLarge deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes
In this paper we prove exact forms of large deviations for local times and intersection local times of fractional Brownian motions and Riemann–Liouville processes. We also show that a fractional Brownian motion and the related Riemann–Liouville process behave like constant multiples of each other with regard to large deviations for their local and intersection local times. As a consequence of o...
متن کاملSmall Deviations of Gaussian Random Fields in Lq–Spaces
We investigate small deviation properties of Gaussian random fields in the space Lq(R N , μ) where μ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby “thin” measures μ, i.e., those which are singular with respect to the N–dimensional Lebesgue measure; the so–called self–similar measures providing a class of typical examples. For a large class of random fi...
متن کاملSmall Deviations of Stable Processes and Entropy of the Associated Random Operators
We investigate the relation between the small deviation problem for a symmetric α-stable random vector in a Banach space and the metric entropy properties of the operator generating it. This generalizes former results due to Li and Linde and to Aurzada. It is shown that this problem is related to the study of the entropy numbers of a certain random operator. In some cases an interesting gap app...
متن کامل