Section: New Results

Local Hölder regularity of Set-Indexed processes

Participants : Erick Herbin, Alexandre Richard.

In the set-indexed framework of Ivanoff and Merzbach ( [62] ), stochastic processes can be indexed not only by $𝐑$ but by a collection $𝒜$ of subsets of a measure and metric space $(𝒯,d,m)$, with some assumptions on $𝒜$. In we introduce and study some assumptions $\left(A_1\right)$ and $\left(A_2\right)$ on the metric indexing collection $(𝒜,d_{𝒜})$ in order to obtain a Kolmogorov criterion for continuous modifications of SI stochastic processes. Under this assumption, the collection is totally bounded and a set-indexed process with good incremental moments will have a modification whose sample paths are almost surely Hölder continuous, for the distance $d_{𝒜}$. Once this condition is established, we investigate the definition of Hölder coefficients for SI processes. We shall denote ${\tilde{\alpha }}_X\left(t\right)$ and ${\alpha }_X\left(t\right)$ for the local and pointwise Hölder exponents of $X$ at $t$, and ${\tilde{\alpha }}_X\left(t\right)$ and ${\alpha }_X\left(t\right)$ for their deterministic counterpart in case $X$ is Gaussian.

In [18] , a set-indexed extension for fractional Brownian motion has been defined and studied. A mean-zero Gaussian process ${𝐁}^H=\left\{{𝐁}_U^H,U\in 𝒜\right\}$ is called a set-indexed fractional Brownian motion (SIfBm for short) on $(𝒯,𝒜,m)$ if

where $H\in (0,1/2]$ is the index of self-similarity of the process.

In [12] , ${\tilde{\alpha }}_X$ and ${\tilde{\alpha }}_X$ have been determined for the particular case of an SIfBm indexed by the collection $\{[0,t];t\in {𝐑}_{+}^N\}\cup \left\{\varnothing \right\}$, called the multiparameter fractional Brownian motion. If $X$ denotes the ${𝐑}_{+}^N$-indexed process defined by $X_t={𝐁}_{[0,t]}^H$ for all $t\in {𝐑}_{+}^N$, it is proved that for all $t_0\in {𝐑}_{+}^N$, ${\tilde{\alpha }}_X\left(t_0\right)=H$ and with probability one, for all $t_0\in {𝐑}_{+}^N$, ${\tilde{\alpha }}_X\left(t_0\right)=H$. A theorem of allows one to extend these results to SIfBm indexed by a more general class than the sole collection of rectangles of ${𝐑}_{+}^N$.

Theorem 0.1 Let ${𝐁}^H$ be a set-indexed fractional Brownian motion on $\left(𝒯,𝒜,m\right)$, $H\in (0,1/2]$. Assume that the subclasses ${\left({𝒜}_n\right)}_{n\in 𝐍}$ satisfy Assumption $\left(A_1\right)$.

Then, the local and pointwise Hölder exponents of ${𝐁}^H$ at any $U_0\in 𝒜$, defined with respect to the distance $d_m$ or any equivalent distance, satisfy

and if Assumption $\left(A_2\right)$ holds,

Consequently, since the collection $𝒜$ of rectangles of ${𝐑}_{+}^N$ with $m$ the Lebesgue measure satisfies $\left(A_1\right)$ and $\left(A_2\right)$, we obtained a new result on a classical multiparameter process: the multiparameter fractional Brownian motion ${𝐁}^H$ satisfy, for $T\in {𝐑}_{+}^N$: