International audienceIn this paper, we study the Hölder regularity of set-indexed stochastic processes defined in the framework of Ivanoff-Merzbach. The first key result is a Hölder-continuity Theorem derived from the approximation of the indexing collection by a nested sequence of finite subcollections. Hölder-continuity based on the increment definition for set-indexed processes is also considered. Then, the localization of these properties leads to various definitions of Hölder exponents. Moreover, a pointwise continuity exponent is defined in relation with the weak continuity property for set-indexed processes which only considers single point jumps. In the case of Gaussian processes, almost sure values are proved for the Hölder exponents. As an application, the local regularity of the set-indexed fractional Brownian motion and the Ornstein-Uhlenbeck process are proved to be constant, with probability one

No abstract

Using structures of Abstract Wiener Spaces, we define a fractional Brownian field indexed by a product space (0, 1/2] × L 2 (T, m), (T, m) a separable measure space, 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 motions and multiparameter fractional Brownian motions, and provides a setup for new ones. We prove that it has satisfactory incremental variance in both coordinates and derive certain continuity and Hölder regularity properties in relation with metric entropy. Also, a sharp estimate of the small ball probabilities is provided, generalizing a result on Lévy fractional Brownian motion. Then, we apply these general results to multiparameter and set-indexed processes, proving the existence of processes with prescribed local Hölder regularity on general indexing collections.

We propose a new approach to obtain quantitative convergence of moderately interacting particle systems to solutions of nonlinear Fokker-Planck equations with singular kernels. Our result only requires very weak regularity on the interaction kernel, including the Biot-Savart kernel, the family of Keller-Segel kernels in arbitrary dimension, and more generally singular Riesz kernels. This seems to be the first time that such quantitative convergence results are obtained in Lebesgue and Sobolev norms for the aforementioned kernels. In particular, this convergence holds locally in time for PDEs exhibiting a blow-up in finite time. The proof is based on a semigroup approach combined with stochastic calculus techniques, and we also exploit the regularity of the solutions of the limiting equation.Furthermore, we obtain well-posedness for the McKean-Vlasov SDEs involving these singular kernels and we prove the trajectorial propagation of chaos for the associated moderately interacting particle systems.