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Optimal Markovian Couplings and Applications

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Acta Mathematica Sinica,English Series 1994年第3期10卷 260-275页

作者： chen mufa department of mathematics beijing normal university beijing,100875 china 1. department of mathematics Beijing Normal University 100875 Beijing China

This paper is devoted to studying a new topic:optimal Markovian couplings,mainly for time-continuous Markov *** study emphasizes the analysis of the coupling opera- tors rather than the *** constructions of optimal Markovian couplings for Markov chains and diffusions are presented,which are often ***,the results are applied to study the L^2-convergence for Markov chains and for a diffusion on compact *** estimate of the convergent rate provided by this method can be sharp.

关键词： chen Optimal Markovian Couplings and Applications OMC

Estimate of Exponential Convergence Rate in Total Variation by Spectral Gap

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Acta Mathematica Sinica,English Series 1998年第1期14卷 9-16页

作者： chen mufa department of mathematics, beijing normal university, beijing 100875, china 北京师范大学数学系北京

This note is devoted to study the exponential convergence rate in the total variation for reversible Markov processes by comparing it with the spectral gap. It is proved that in a quite general setup, with a suitable restriction on the initial distributions, the rate is bounded from below by the spectral gap. Furthemore, in the compact case or for birth-death processes or half-line diffosions, the rate is shown to be equal to the spectral gap.

关键词： Total variation Spectral gap Markov processes

Estimation of Spectral Gap for Markov Chains

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Acta Mathematica Sinica,English Series 1996年第4期12卷 337-360页

作者： chen mufa department of mathematics beijing normal university beijing, 100875 china department of mathematics Beijing Normal University Beijing China

The study of the convergent rate (spectra gap) in the L^2-sense is motivated from several different fields: probability statistics, mathematical physics, computer science and so on and it is now an artive research topic. Based on a new approach (the coupling technique) introduced in [7] for the estimate of the convergent rate and as a continuation of [4],[5],[7-9],[23] and [24], this paper studies the estimate of the rate for time-continuous Markov chains. Two variational formulas for the rate are presented here for the first time for birth-death processes. For diffusions, similar results are presented in an accompany paper [10]. The new formulas enable us to recover or improve the main known results. The connection between the sharp estimate and the corresponding eigenfunction is explored and illustrated by various examples. A previous result on optimal Markovian couplings is also extended in the paper.

关键词： Markov chains Spectral gap Couplings

The Principal Eigenvalue for Jump Processes

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Acta Mathematica Sinica,English Series 2000年第3期16卷 361-368页

作者： mufa chen department of mathematics,beijing normal university,beijing 100875,P.R.china department of mathematics Beijing Normal University Beijing 100875 P. R. China

A variational formula for the lower bound of the principal eigenvalue of general Markov jump processes is *** result is complete in the sense that the condition is fulfilled and the resulting bound is sharp for Markov... 详细信息

关键词： Principal eigenvalue Jump processes Variational formula for Dirichlet form

Nash Inequalities for General Symmetric Forms

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Acta Mathematica Sinica,English Series 1999年第3期15卷 353-370页

作者： mufa chen department of mathematics, beijing normal university, beijing 100875, P. R. china department of mathematics Beijing Normal University Beijing P. R. China

This paper deals with the Nash inequalities and the related ones for general symmetric forms which can be very much unbounded. Some sufficient conditions in terms of the isoperimetric inequalities and some necessary conditions for the inequalities are presented. The resulting conditions can be sharp qualitatively as illustrated by some examples. It turns out that for a probability measure, the Nash inequalities are much stronger than the Poincare and the logarithmic Sobolev inequalities in the present context.

关键词： Nash inequality Symmetric form Isoperimetric Inequality Jump process

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Saturated models

Chinese Science Bulletin 1998年第7期43卷 562-564页

作者： chen Guolong department of mathematics, beijing normal university, beijing 100875, china department of mathematics Beijing Normal University Beijing China

An equivalent description of saturated models is given, and some properties of countable saturated models, almost saturated models and full models are discussed. New proofs of two important theorems of the related doc... 详细信息

关键词： almost saturated model full model power type.

Reaction-diffusion processes

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Chinese Science Bulletin 1998年第17期43卷 1409-1420页

作者： chen mufa beijing normal university beijing 100875, china beijing normal university Beijing China

The main progress made in the past dozen years or more in the study of reaction-diffusion (abbrev. RD) processes is surveyed. The processes are motivated from some typical models in the modern non-equilibrium statistical physics and consist of an important class of interacting particle systems which is currently an active research field in probability and mathematical physics. The models are concrete but as a part of the infinite-dimensional mathematics, the topic is quite hard. It is explained how new problems arise, and how some new ideas and new mathematical tools are introduced. Surprisingly, the mathematical tools produced from the study on the simple models then turn to have a lot of powerful applications not only in probability theory but also in other branches of mathematics. Nevertheless, the story is still far from finished, and some important open problems are proposed for further study.

关键词： Reaction-diffusion process ergodicity phase transitions coupling.

Recovery of functions via their irregular sampling values

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Chinese Science Bulletin 1998年第4期43卷 268-271页

作者： FANG Gensun department of mathematics, beijing normal university, beijing 100875, china department of mathematics Lkijing Normal University Beijing China

It is proved that if a function can be completely reconstructed by tempered splines which satisfy the real Kiesz basis condition as the degree of spline goes to infinity, then the function is band limited.

关键词： band limited function cardinal splines irregular sampling.

Functional central limit theorem for super α-stable processes

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Science china mathematics 2004年第6期47卷 874-881页

作者： HONG Wenmingdepartment of mathematics, beijing normal university, beijing 100875, china 1. department of mathematics Beijing Normal University 100875 Beijing China

A functional central limit theorem is proved for the centered occupation time process of the super α-stable processes in the finite dimensional distribution sense. For the intermediate dimensions α < d < 2α (0 < α ≤ 2), the limiting process is a Gaussian process, whose covariance is specified; for the critical dimension d= 2α and higher dimensions d < 2α, the limiting process is Brownian motion.

关键词： super α-stable processes occupation time central limit theorem evolution equation