Hydrodynamic limits for the Boltzmann process

We study the behavior of the nonlinear Markov process associated to the Boltzmann equation under both hyperbolic and parabolic space-time scalings. In the first case the limit of the process is the solution of an o.d.e. with vector field given by a solution of the Euler equation, while in the second case the limit of the process, in the incompressible case, turns out to be a diffusion process whose drift is a solution of the incompressible Navier-Stokes equation.     

A mean-field limit for a class of queueing networks

A model of centralized symmetric message-switched networks is considered, where the messages having a common address must be served in the central node in the order which corresponds to their epochs of arrival to the network. The limitN is discussed, whereN is the branching number of the network graph. This procedure is inspired by an analogy with statistical mechanics (the mean-field approximation). The corresponding limit theorems are established and the limiting probability distribution for the network response time is obtained. Properties of this distribution are discussed in terms of an associated boundary problem.     

离散时间单死链的可加泛函北大核心CSCD

本文给出了离散单死链的可加泛函高阶矩的显式递推表达,并研究了离散单死链的多项式遍历和中心极限定理.     

具有多重极限环的微分系统的近似系统

本文利用Ｈａｍｉｔｏｌｎ系统的扰动系统得到了构造具有多重极限环的微分系统的近似系统的一种系统并举出了一些具体例子。     

Randomly sampled Riemann sums and complete convergence in the law of large numbers for a case without identical distribution

Let the points be independently and uniformly randomly chosen in the intervals , where . We show that for a finite-valued measurable function on , the randomly sampled Riemann sums converge almost surely to a finite number as if and only if , in which case the limit must agree with the Lebesgue integral. One direction of the proof uses Bikelis' (1966) non-uniform estimate of the rate of convergence in the central limit theorem. We also generalize the notion of sums of i.i.d. random variables, subsuming the randomly sampled Riemann sums above, and we show that a result of Hsu, Robbins and Erd\H{o}s (1947, 1949) on complete convergence in the law of large numbers continues to hold. In the Appendix, we note that a theorem due to Baum and Katz (1965) on the rate of convergence in the law of large numbers also generalizes to our case.

     

A Wolff-Denjoy theorem for infinitely connected Riemann surfaces

We generalize the classical Wolff-Denjoy theorem to certain infinitely connected Riemann surfaces. Let be a non-parabolic Riemann surface with Martin boundary . Suppose each Martin function , , extends continuously to and vanishes there. We show that if is an endomorphism of and the iterates of converge to the point at infinity, then the iterates converge locally uniformly to a point in . As an application, we extend the Wolff-Denjoy theorem to non-elementary Gromov hyperbolic covering spaces of compact Riemann surfaces. Such covering surfaces are of independent interest. Finally, we use the theory of non-tangential boundary limits to give a version of the Wolff-Denjoy theorem that imposes certain mild restrictions on but none on itself.

     

Central limit theorem and infinitely divisible probabilities associated with partial differential operators

We provide

Subcritical asymptotic behavior in the thermodynamic limit of reversible random polymerization processes

We consider a reversible Markov process as a chemical polymerization model and study the asymptotic behavior (in the thermodynamic limit asN+) of a particular probability distribution on the set ofN-dimensional vectors, thekth component of which is the number ofk-mers. The study establishes the existence of three stages (subcritical, near-critical, and supercritical stages) of polymerization, depending on the value of the strength of the fragmentation reaction. The present paper concentrates on the analysis of the subcritical stage. In the subcritical stages we show that the size of the largest length of polymers of sizeN is of the order logN asN+.     

Sharp Conditions for the CLT of Linear Processes in a Hilbert Space

In this paper we study the behavior of sums of a linear process associated to a strictly stationary sequence with values in a real separable Hilbert space and are linear operators from H to H. One of the results is that satisfies the CLT provided are i.i.d. centered having finite second moments and . We shall provide an example which shows that the condition on the operators is essentially sharp. Extensions of this result are given for sequences of weak dependent random variables under minimal conditions.     

Renormalizing rectangles and other topics in random matrix theory

We consider random Hermitian matrices made of complex or realM×N rectangular blocks, where the blocks are drawn from various ensembles. These matrices haveN pairs of opposite real nonvanishing eigenvalues, as well asM–N zero eigenvalues (forM>N). These zero eigenvalues are kinematical in the sense that they are independent of randomness. We study the eigenvalue distribution of these matrices to leading order in the large-N, M limit in which the rectangularityr=M/N is held fixed. We apply a variety of methods in our study. We study Gaussian ensembles by a simple diagrammatic method, by the Dyson gas approach, and by a generalization of the Kazakov method. These methods make use of the invariance of such ensembles under the action of symmetry groups. The more complicated Wigner ensemble, which does not enjoy such symmetry properties, is studied by large-N renormalization techniques. In addition to the kinematical -function spike in the eigenvalue density which corresponds to zero eigenvalues, we find for both types of ensembles that if |r–1| is held fixed asN, theN nonzero eigenvalues give rise to two separated lobes that are located symmetrically with respect to the origin. This separation arises because the nonzero eigenvalues are repelled macroscopically from the origin. Finally, we study the oscillatory behavior of the eigenvalue distribution near the endpoints of the lobes, a behavior governed by Airy functions. Asr1 the lobes come closer, and the Airy oscillatory behavior near the endpoints that are close to zero breaks down. We interpret this breakdown as a signal thatr1 drives a crossover to the oscillation governed by Bessel functions near the origin for matrices made of square blocks.