## Section: New Results

### Hölder regularity of Set-Indexed processes

Participants : Erick Herbin, Alexandre Richard.

*In collaboration with Prof. Ely Merzbach (Bar Ilan University, Israel).*

In the set-indexed framework of Ivanoff and Merzbach ( [54] ), stochastic processes can be indexed not only by $\mathbb{R}$ 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 [25] , we introduce and study some assumptions 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. From the real-parameter case, the most straightforward are the local (and pointwise) Hölder exponents around ${U}_{0}\in \mathcal{A}$:

When the processes are Gaussian, a deterministic counterpart to this exponent is defined as it is in the real-parameter framework. For all ${U}_{0}\in \mathcal{A}$, we proved that almost surely, the random and the deterministic exponents are equal. Also, we proved that for the local exponents, this result holds almost surely, uniformly on $\mathcal{A}$.

Given the particular structure of $\mathcal{A}$, other coefficients of Hölder regularity were studied on $\mathcal{C}$:

$\mathcal{C}=\left\{A\setminus \bigcup _{k=1}^{n}{B}_{k}:A,{B}_{1},\cdots ,{B}_{n}\in \mathcal{A},n\in \mathbb{N}\right\}.$ | (15) |

On specific subclasses ${\mathcal{C}}^{l}$ of $\mathcal{C}$ (satisfying ${\cup}_{l\ge 1}{\mathcal{C}}^{l}=\mathcal{C}$), the local (and pointwise) ${\mathcal{C}}^{l}$-Hölder exponents are defined:

${\tilde{\alpha}}_{X,{\mathcal{C}}^{l}}\left({U}_{0}\right)=sup\left\{\alpha :\underset{\rho \to 0}{lim\; sup}\underset{\begin{array}{c}U\in {B}_{{d}_{\mathcal{A}}}({U}_{0},\rho )\\ V\in {\mathcal{B}}^{l}({U}_{0},\rho )\end{array}}{sup}\frac{|\Delta {X}_{U\setminus V}|}{{d}_{\mathcal{A}}{(U,V)}^{\alpha}}<\infty \right\},$ | (16) |

and this definition is proved to be independent of $l$, leading to the definition of ${\tilde{\alpha}}_{X,\mathcal{C}}\left({U}_{0}\right)$. It is compared to ${\tilde{\alpha}}_{X}\left({U}_{0}\right)$ and related to the Hölder exponent of the process projected on flows (a flow is a continuous increasing path in $\mathcal{A}$). This last technique permits to show that the pointwise Hölder exponent of the SIfBm is almost surely uniformly equal to $H$, the Hurst parameter of the SIfBm. This completes some previous results on the multiparameter fractional Brownian motion.

The last exponent which is studied is the exponent of pointwise continuity:

${\alpha}_{X}^{pc}\left(t\right)=sup\left\{\alpha :\phantom{\rule{4pt}{0ex}}\underset{n\to \infty}{lim\; sup}\frac{|\Delta {X}_{{C}_{n}\left(t\right)}|}{m{\left({C}_{n}\left(t\right)\right)}^{\alpha}}<\infty \right\}$ | (17) |

for all $t\in \mathcal{T}$, where ${C}_{n}\left(t\right)$ is the smaller set of ${\mathcal{C}}_{n}$ containing $t$. Almost sure results are also obtained in that case. For instance, the coefficient of pointwise continuity of a SI Brownian motion equals $1/2\phantom{\rule{4pt}{0ex}}a.s.$

All these results are finally applied to the SIfBm and the SI Ornstein-Ühlenbeck process ([1] ).