## 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 $\mathbf{R}$ but by a collection $\mathcal{A}$ of subsets of a measure and metric space $(\mathcal{T},d,m)$, with some assumptions on $\mathcal{A}$. In we introduce and study some assumptions $\left({A}_{1}\right)$ and $\left({A}_{2}\right)$ on the metric indexing collection $(\mathcal{A},{d}_{\mathcal{A}})$ 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 ${d}_{\mathcal{A}}$. 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 ${\mathbf{B}}^{H}=\left\{{\mathbf{B}}_{U}^{H},U\in \mathcal{A}\right\}$ is called a *set-indexed fractional Brownian motion (SIfBm for short)* on $(\mathcal{T},\mathcal{A},m)$ if

$\forall U,V\in \mathcal{A},\phantom{\rule{1.em}{0ex}}\mathbf{E}\left[{\mathbf{B}}_{U}^{H}{\mathbf{B}}_{V}^{H}\right]=\frac{1}{2}\left[m{\left(U\right)}^{2H}+m{\left(V\right)}^{2H}-m{(U\u25b3V)}^{2H}\right],$ | (6) |

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 $\{[0,t];\phantom{\rule{0.277778em}{0ex}}t\in {\mathbf{R}}_{+}^{N}\}\cup \left\{\varnothing \right\}$, called the *multiparameter fractional Brownian motion*.
If $X$ denotes the ${\mathbf{R}}_{+}^{N}$-indexed process defined by ${X}_{t}={\mathbf{B}}_{[0,t]}^{H}$ for all $t\in {\mathbf{R}}_{+}^{N}$, it is proved that for all ${t}_{0}\in {\mathbf{R}}_{+}^{N}$, ${\tilde{\alpha}}_{X}\left({t}_{0}\right)=H$ and with probability one, for all ${t}_{0}\in {\mathbf{R}}_{+}^{N}$, ${\tilde{\alpha}}_{X}\left({t}_{0}\right)=H$.
A theorem of allows one to extend these results to SIfBm indexed by a more general class than the sole collection of rectangles of ${\mathbf{R}}_{+}^{N}$.

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

Then, the local and pointwise Hölder exponents of ${\mathbf{B}}^{H}$ at any ${U}_{0}\in \mathcal{A}$, 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 $\mathcal{A}$ of rectangles of ${\mathbf{R}}_{+}^{N}$ 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 ${\mathbf{B}}^{H}$ satisfy, for $T\in {\mathbf{R}}_{+}^{N}$: