Some Singular Sample Path Properties of a Multiparameter Fractional Brownian Motion

Abstract

We obtain a spectral representation and compute the small ball probabilities for a (non-increment stationary) multiparameter extension of the fractional Brownian motion. We derive from these results a Chung-type law of the iterated logarithm at the origin, and exhibit the singular behaviour of this multiparameter fractional Brownian motion, as it behaves very differently at the origin and away from the axes. A functional version of this Chung-type law is also provided.

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

“…We will need it to be Hilbert-Schmidt for an extension of Bochner's theorem to be valid. This is the content of the following lemma, proved in [14] (actually with slightly stronger conclusions than written here). Some Hilbert spaces associated to a kernel (as for instance a covariance) will be particularly useful.…”

“…On the contrary to the Lévy fBm and the fractional Brownian sheet, the spectral representation for this process is only recent. In [14], it was obtained as a special case of our theorem, due to special results available for stable measures on Hilbert spaces. Hence the present work yields a more generic and complete (although more lengthy) way to prove that:…”

“…where E is some Hilbert space in which L 2 ( d + ) is (Hilbert-Schmidt) embedded, γ is defined as in Theorem 2.1, and M H has control measure ∆ H which is the Lévy measure of a stable measure on E. In particular, this representation has applications on the sample paths of the multiparameter fBm, since B H t can now be written as a sum of independent processes if E is sliced into disjoint subsets [14]. It is also interesting to notice that ∆ H has a similar form to the control measure of the usual fractional Brownian motion.…”

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.…”

Local Hölder regularity for set-indexed processes

“…Example 2.10. In [37], it is explained how to construct a stable measure on an abstract Hilbert space E such that:…”