TY - JOUR

T1 - Small deviations of gaussian random fields in lq–spaces

AU - Lifshits, Mikhail

AU - Linde, Werner

AU - Shi, Zhan

PY - 2006/1/1

Y1 - 2006/1/1

N2 - We investigate small deviation properties of Gaussian random fields in the space Lq(ℝN, 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 Lq(ℝN, 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.

AB - We investigate small deviation properties of Gaussian random fields in the space Lq(ℝN, 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 Lq(ℝN, 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.

KW - Fractal measures

KW - Fractional Brownian motion

KW - Gaussian processes

KW - Hölder operators

KW - Kolmogorov numbers

KW - Metric entropy

KW - Random fields

KW - Self–similar measures

KW - Small deviations

UR - http://www.scopus.com/inward/record.url?scp=33845425130&partnerID=8YFLogxK

U2 - 10.1214/EJP.v11-379

DO - 10.1214/EJP.v11-379

M3 - Article

AN - SCOPUS:33845425130

VL - 11

SP - 1204

EP - 1233

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

SN - 1083-6489

ER -