2006/05/16 by Mikhail Lifshits, Lifshits, Mikhail, Werner Linde +3
Economics, Econometrics and Finance · Environmental Science · Mathematics · #60G15 #60G18 #Analysis of environmental and stochastic processes #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #math.PR #msc:60G15 #msc:60G18
paper · pdf · doi:10.48550/arxiv.math/0605417
arxiv created 2006/05/16 · openalex publication_date 2006/05/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.