Local Fields, Gaussian Measures, and Brownian Motions

ar X iv : m at h / 98 03 04 6 v 1 [ m at h . PR ] 1 1 M ar 1 99 8 LOCAL FIELDS , GAUSSIAN MEASURES , AND BROWNIAN MOTIONS Steven

Small Deviations of Gaussian Random Fields in Lq–Spaces

We investigate small deviation properties of Gaussian random fields in the space Lq(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 fi...

Fractal Measures of the Sets Associated to Gaussian Random Fields

This paper summarizes recent results about the Hausdorff measure of the image, graph and level sets of Gaussian random fields, the packing dimension and packing measure of the image of fractional Brownian motion, the local times and multiple points of Gaussian random fields. Some open problems are also pointed out.

Large deviations for the local and intersection local times of fractional Brownian motions

Large deviation principle for the non-linear functionals of non-Markovian models is a challenging subject. A class of such models are Gaussian processes. Among them, the fractional Brownian motions are perhaps the most important processes. In this talk, I will talk about some recent progress achieved in the large deviations for local times and intersection local times of fractional Brownian mot...

Continuity in the Hurst Index of the Local Times of Anisotropic Gaussian Random Fields

Let {{XH(t), t ∈ RN}, H ∈ (0, 1)N} be a family of (N, d)-anisotropic Gaussian random fields with generalized Hurst indices H = (H1, . . . ,HN ) ∈ (0, 1) . Under certain general conditions, we prove that the local time of {XH0(t), t ∈ RN} is jointly continuous whenever ∑N `=1 1/H 0 ` > d. Moreover we show that, when H approaches H , the law of the local times of X(t) converges weakly [in the spa...