Section: New Results

A fractional Brownian field indexed by $L^2$ and a varying Hurst parameter

Participant : Alexandre Richard.

Using structures of Abstract Wiener Spaces and their reproducing kernel Hilbert spaces, we define a fractional Brownian field indexed by a product space $(0,1/2]\times L^2(T,m)$, 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 motion and multiparameter fractional Brownian motion, and provides a setup for new ones. We prove that it has good incremental variance in both coordinates and derive certain continuity and Hölder regularity properties. Then, we apply these general results to multiparameter and set-indexed processes, which proves the existence of processes with prescribed local Hölder regularity on general indexing collections.

The family of fBm can be considered for the different Hurst parameters as a single Gaussian process indexed by $(h,t)\in (0,1)\times {ℝ}_{+}$, which is the position we adopt. Besides, the “time” indexing is replaced by any separable $L^2$ space. We prove that there exists a Gaussian process indexed by $(0,1/2]\times L^2(T,m)$, with the additional constraint that the variance of its increments is as well behaved as it was on $(0,1)\times {ℝ}_{+}$, that is, for any compact of $L^2$, there is a constant $C>0$ such that for any $f$ in this compact, and any $h,h^{\text{'}}\in (0,1/2)$,

When looking at the $L^2$–fBf with a fixed $h$, we have the following covariance: for each $h\in (0,1/2]$,

An important subclass of these processes is formed by processes restricted to indicator functions of subsets of $T$. In particular, multiparameter when $(T,m)=({ℝ}_{+}^d,\mathrm{Leb}.)$, and more largely set-indexed processes [62] ,[20] naturally appear and thus motivate generalization b), besides the inherent interest of studying processes over an abstract space.

To define this field, we used fractional operators on the Wiener space $W$ introduced in [56] , and first expressed the fractional Brownian field (indexed by $(0,1/2]\times {ℝ}_{+}$) as a white noise integral over $W$:

The advantage of this approach is to allow the transfer of techniques of calculus on the Wiener space to any other linearly isometric space with the same structure (those spaces are called Abstract Wiener Spaces). Using the separability and reproducing kernel property of the Cameron-Martin spaces built from the kernels $k_h,h\in (0,1/2]$, we prove the existence of a Brownian field $\{{𝐁}_{h,f},h\in (0,1/2],f\in L^2(T,m)\}$ over some probability space $(\Omega ,ℱ,\mathbb{P})$. Some Hilbert space analysis then provides the desired bound (22 ). Then, we used this to derive a sufficient condition for almost sure continuity of the fractional Brownian field, in terms of metric entropy.

For fixed $h$, we proved that the $h$-fractional Brownian motion has the strong local nondeterminism property, which allowed to compute a sharp estimate of its small deviations, that is, for a compact $K$ of $L^2$:

where $N(K,d_h,\epsilon )$ is the metric entropy of $K$, i.e., the minimal number of balls necessary to cover $K$ with $d_h$-balls (the metric induced by the $h$-fBm) of radius at most $\epsilon$.

Finally, we looked at the Hölder regularity of the fBf, when the $L^2$ indexing collection is restricted to the indicator functions of the rectangles of ${ℝ}^d$ (multiparameter processes) or to some indexing collection (in the sense of [62] ). This restriction permits to use local Hölder regularity exponents, in the flavour of what was done in [24] . When a regular path $𝐡:L^2\to (0,1/2]$ is specified, this defines a multifractional Brownian field as ${𝐁}_f^{𝐡}={𝐁}_{𝐡\left(f\right),f}$, whose Hölder regularity at each point is proved to equal $𝐡\left(f\right)$ almost surely.