A fractional Brownian field indexed byL2and a varying Hurst parameter

Abstract

Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space (0, 1/2] × L 2 (T, m), (T, m) a separable measure space, where the first coordinate corresponds to the Hurst parameter of fractional Brownian motion. This field encompasses a large class of existing fractional Brownian processes, such as Lévy fractional Brownian motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Hölder regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on Lévy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local Hölder regularity on general indexing collections.

Stable Processes with Stationary Increments Parameterized by Metric Spaces

“…. , t n ) ∈ R n + , in which case the process is referred to as a multiparameter fractional Brownian motion (see also [31,32]). This process has different properties from other extensions of fractional Brownian motions, and in particular it does not have stationary increments (see e.g.…”

“…This process has different properties from other extensions of fractional Brownian motions, and in particular it does not have stationary increments (see e.g. [31] for comparisons). Following the same notion, we refer to We next consider a natural decomposition of set-indexed fractional Brownian motions Y 2,β , inspired by the decomposition of fractional Brownian motions by bifractional Brownian motions introduced by Lei and Nualart [23].…”

Some Singular Sample Path Properties of a Multiparameter Fractional Brownian Motion

“…Now, we address the spectral decomposition itself. For any α ∈ (0, 2], the application ξ ∈ E * → Sξ α H is continuous (because of the inequality · H ≤ C · E * ) and negative definite (by an argument on Bernstein functions, see for instance the introduction of [27]). Thus, according to Schoenberg's theorem, ξ → exp(−t Sξ α H ) is positive definite for any t ∈ * + .…”

“…where λ denotes the Lebesgue measure in ν , [0, t] is the rectangle with vertices at 0 and t, and is the symmetric difference of sets. This is a special case of a family of covariance on sets introduced by Herbin and Merzbach [13] to define the set-indexed fractional Brownian motion, and extended in [27] to a more general expression as a covariance on L 2 (T, m). This process differs from the other extensions that are the Lévy fractional Brownian motion, and the fractional Brownian sheet, although it shares several properties with them (see [27] for a more thorough discussion on the links between these processes).…”

Increment stationarity of $L^2$-indexed stochastic processes: spectral representation and characterization

“…However there are examples where the theorem permits to deduce sample path properties of multiparameter processes [14]. In all this section, the prototypical example of a process to which our spectral representation theorem applies is the L 2 (T, m)-indexed fractional Brownian motion (defined in [13] as an extension of the set-indexed fractional Brownian motion [5]). Hence in Section 3 of this paper, we focus on the L 2 (T, m)-indexed fBm.…”