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 but by a collection of subsets of a measure and metric space , with some assumptions on . In [25] , we introduce and study some assumptions on the metric indexing collection 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 .
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 :
When the processes are Gaussian, a deterministic counterpart to this exponent is defined as it is in the real-parameter framework. For all , 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 .
Given the particular structure of , other coefficients of Hölder regularity were studied on :
On specific subclasses of (satisfying ), the local (and pointwise) -Hölder exponents are defined:
and this definition is proved to be independent of , leading to the definition of . It is compared to and related to the Hölder exponent of the process projected on flows (a flow is a continuous increasing path in ). This last technique permits to show that the pointwise Hölder exponent of the SIfBm is almost surely uniformly equal to , 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:
for all , where is the smaller set of containing . Almost sure results are also obtained in that case. For instance, the coefficient of pointwise continuity of a SI Brownian motion equals
All these results are finally applied to the SIfBm and the SI Ornstein-Ühlenbeck process ([1] ).