Kolmogorov numbers of Riemann–Liouville operators over small sets and applications to Gaussian processes

AbstractWe 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 entropy numbers of K. Under some regularity assumption about the entropy of K these estimates turn out to be two-sided. By standard methods the results are also valid for the (dyadic) entropy numbers of Rα. Finally, we apply these estimates for the investigation of the small ball behavior of certain Gaussian stochastic processes, as e.g. fractional Brownian motion or Riemann–Liouville processes, indexed by small (fractal) sets

Small Deviations of Gaussian Random Fields in LqL_q--Spaces

We investigate small deviation properties of Gaussian random fields in the space $L_q(\R^N,\mu)$ where $\mu$ is an arbitrary finite compactly supported Borel measure. Of special interest are hereby "thin" measures $\mu$, 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 fields (including, among others, fractional Brownian motions), we describe the behavior of small deviation probabilities via numerical characteristics of $\mu$, called mixed entropy, characterizing size and regularity of $\mu$. For the particularly interesting case of self--similar measures $\mu$, the asymptotic behavior of the mixed entropy is evaluated explicitly. As a consequence, we get the asymptotic of the small deviation for $N$--parameter fractional Brownian motions with respect to $L_q(\R^N,\mu)$--norms. While the upper estimates for the small deviation probabilities are proved by purely probabilistic methods, the lower bounds are established by analytic tools concerning Kolmogorov and entropy numbers of H\"older operators

Functional quantization and metric entropy for Riemann-Liouville processes

We derive a high-resolution formula for the $L^2$-quantization errors of Riemann-Liouville processes and the sharp Kolmogorov entropy asymptotics for related Sobolev balls. We describe a quantization procedure which leads to asymptotically optimal functional quantizers. Regular variation of the eigenvalues of the covariance operator plays a crucial role

Mini-Workshop: Small Deviation Problems for Stochastic Processes and Related Topics

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Liouville Quantum Gravity on the unit disk

Our purpose is to pursue the rigorous construction of Liouville Quantum Field Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two authors in the context of the Riemann sphere and inspired by the 1981 seminal work by Polyakov. In this paper, we investigate the case of simply connected domains with boundary. We also make precise conjectures about the relationship of this theory to scaling limits of random planar maps with boundary conformally embedded onto the disk