Small Deviations of Gaussian Random Fields in $L_q$--Spaces

Abstract

We investigate small deviation properties of Gaussian random fields in the space $L_q(\R^N,μ)$ 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 fields (including, among others, fractional Brownian motions), we describe the behavior of small deviation probabilities via numerical characteristics of $μ$, called mixed entropy, characterizing size and regularity of $μ$. For the particularly interesting case of self--similar measures $μ$, 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 $L_q(\R^N,μ)$--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 Hölder operators.