Recent Developments on Fractal Properties of Gaussian Random Fields

Packing dimensions of the images of Gaussian random fields

“…We refer to Taylor (1986) and Xiao (2004) 3 for surveys on fractal properties of Markov processes, and to Adler (1981), Kahane (1985) and Xiao ( , 2013 for results 4 on Gaussian random fields. 5 Let X = {X(t), t ∈ R N } be a Gaussian random field with values in R d defined on a probability space (Ω, F , P) by 6 X (t) = (X 1 (t), .…”

Stable Processes with Stationary Increments Parameterized by Metric Spaces

“…Recent developments on fractional Lévy Brownian fields include for example [7,21] (indexed by S n ). More generally, in the spatial context a Gaussian process {G x } x∈M is often characterized by its variogram v(x, y) := E(G x − G y ) 2 (then stationary increments imply that v(x, y) is a function of d(x, y)), and there is already a huge literature on random fields from this aspect; see for example [3,42] for latest surveys on Gaussian random fields. Other types of generalizations of fractional Brownian motions include for example [15,27].…”

Strong local nondeterminism of spherical fractional Brownian motion

“…For studying these and many other problems on Gaussian random fields, the appropriate properties of strong local nondeterminism (SLND) have proven to be more powerful. Instead of recalling definitions of various forms of strong local nondeterminism for (isotropic or anisotropic) Gaussian random fields indexed by R N and their applications, we refer to Xiao [26,27,28] for more information.…”