Empirical and Gaussian processes on Besov classes

算子理论中的Besov空间

Besov spaces in operator theory

本综述探讨了Besov类在算子理论中的多种应用，包括描述Schatten-von Neumann类的Hankel算子、Hilbert空间上幂有界算子多项式的范数估计，以及在算子扰动理论中的重要作用。

Besov regularity of the uniform empirical process

The paths of Brownian motion have been widely studied in the recent years relatively in Besov spaces $B_{p, \infty}^\a$. The results are the same as to the Brownian bridge. In fact these regularities properties are established in some sequence spaces $S_{p, \infty}^\a$ using an isomorphisim between them and $B_{p, \infty}^\a$. In this note, we are concerned with the regularity of the paths of the continuous version of the uniform empirical process in the space $S_{p, \infty}^\a$ and in one of his separable sub space $S_{p, \infty}^{\a, 0}$ for a suitable choice of $\a$ and $p$.

Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities

We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. As a highlight of the paper, we obtain a far reaching $L^p$-analogue, $p \ge 1$, of the Sobolev inequality that was proved for $p=2$ by N. Varopoulos under the assumption of ultracontractivity for the heat semigroup. The case $p=1$ is of special interest since it yields isoperimetric type inequalities.

BV functions and fractional Laplacians on Dirichlet spaces

We study $L^p$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. The main tool is the theory of heat semigroup based Besov classes in Dirichlet spaces that was introduced by the authors in previous works.

Large-time behavior of finite-state mean-field systems with multi-classes

We study in this paper the large-time asymptotics of the empirical vector associated with a family of finite-state mean-field systems with multi-classes. The empirical vector is composed of local empirical measures characterizing the different classes within the system. As the number of particles in the system goes to infinity, the empirical vector process converges towards the solution to a McKean-Vlasov system. First, we investigate the large deviations principles of the invariant distribution from the limiting McKean-Vlasov system. Then, we examine the metastable phenomena arising at a larg…

A weak convergence result for sequential empirical processes under weak dependence

The purpose of this paper is to prove a weak convergence result for empirical processes indexed in general classes of functions and with an underlying $α$-mixing sequence of random variables. In particular the uniformly boundedness assumption on the function class, which is required in most of the existing literature, is spared. Furthermore under strict stationarity a weak convergence result for the sequential empirical process indexed in function classes is obtained, as a direct consequence. Two examples in mathematical statistics, that cannot be treated with existing results, are given as po…