## Electronic Journal of Probability

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

#### Abstract

We investigate small deviation properties of Gaussian random fields in the space $L_q(R^N,\mu)$ 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 Holder operators.

#### Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 46, 1204-1233.

Dates
Accepted: 8 December 2006
First available in Project Euclid: 31 May 2016

https://projecteuclid.org/euclid.ejp/1464730581

Digital Object Identifier
doi:10.1214/EJP.v11-379

Mathematical Reviews number (MathSciNet)
MR2268543

Zentralblatt MATH identifier
1127.60030

Rights

#### Citation

Lifshits, Mikhail; Linde, Werner; Shi, Zhan. Small Deviations of Gaussian Random Fields in $L_q$-Spaces. Electron. J. Probab. 11 (2006), paper no. 46, 1204--1233. doi:10.1214/EJP.v11-379. https://projecteuclid.org/euclid.ejp/1464730581

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