Mu-Fa Chen

From Markov chains to non-equilibrium particle systems

This volume presents a representative work of Chinese probabilists on probability theory and its applications in physics. Interesting results of jump Markov processes are discussed, as well as Markov interacting processes with noncompact states, including the Schlogal model taken from statistical physics. The main body of this book is self-contained and can be used in a course in stochastic processes for graduate students. The book consists of four parts. In Parts 1 and 2, the author introduces the general theory for jump processes. New contributions to the classical problems: uniqueness, recurrence and positive recurrence are studied. Then, probability metrics and coupling methods, stochastically monotonicity, reversibility, large deviations and the estimates of L squared-spectral gap are discussed. Part 3 begins with the study of equilibrium particle systems. This contains the criteria of the reversibility, the construction of Gibbs states and the particle systems on lattice fractals. The final part emphasizes the reaction-diffusion processes which come from non-equilibrium statistical physics. Topics include constructions, existence of stationary distributions, ergodicity, phase transitions and hydrodynamic limits for the processes.

Eigenvalues, Inequalities and Ergodic Theory

This paper surveys the main results obtained during the period 1992—1999 on three aspects mentioned in the title. The first result is a new and general variational formula for the lower bound of spectral gap (i.e. the first non-trivial eigenvalue) of elliptic operators in Euclidean space, Laplacian on Riemannian manifolds or Markov chains (§ 1). Here, a probabilistic method — coupling method is adopted. The new formula is a dual of the classical variational formula. The last formula is actually equivalent to Poincaré inequality. To which, there are closely related logarithmic Sobolev inequality, Nash inequality, Liggett inequality and so on. These inequalities are treated in a unified way by using Cheeger’s method which comes from Riemannian geometry. This consists of § 2. The results on these two aspects are mainly completed by the author joint with F. Y. Wang. Furthermore, a diagram of the inequalities and the traditional three types of ergodicity is presented (§ 3). The diagram extends the ergodic theory of Markov processes. The details of the methods used in the paper will be explained in a subsequent paper under the same title.

General formula for lower bound of the first eigenvalue on Riemannian manifolds

A general formula for the lower bound of the first eigenvalue on compact Riemannian manifolds is presented. The formula improves the main known sharp estimates including Lichnerowicz’ s estimate and Zhong-Yang’s estimate. Moreover, the results are extended to the noncompact manifolds. The study is based on the probabilistic approach (i.e. the coupling method).

Cheeger’s inequalities for general symmetric forms and existence criteria for spectral gap

The Cheeger’s inequalities and some existence criteria for spectral gap and for general symmetric forms are established. The criteria are also extended to general reversible Markov processes but not reported here. Even though in the past several decades, the topics have been widely studied, as far as we know the first problem in the unbounded case and the second one in the general case remain open.

Analytic proof of dual variational formula for the first eigenvalue in dimension one

The first non-zero eigenvalue is the leading term in the spectrum of a self-adjoint operator. It plays a critical role in various applications and is treated in a large number of textbooks. There is a well-known variational formula for it (called the Min-Max Principle) which is especially effective for an upper bound of the eigenvalue. However, for the lower bound of the spectral gap, some dual variational formulas have been obtained only very recently. The original proofs are probabilistic. Some analytic proofs in one-dimensional case are proposed and certain extension is made.

Trilogy of Couplings and General Formulas for Lower Bound of Spectral Gap

This paper starts from a nice application of the coupling method to a traditional topic: the estimation of the spectral gap (=the first non-trivial eigenvalue). Some new variational formulas for the lower bound of the spectral gap of Laplacian on manifold or elliptic operators in Rd or Markov chains are reported [10],[15],[16]. The new formulas are especially powerful for the lower bounds; they have no common points with the classical variational formula (which goes back to Lord Rayleigh (1877) or E. Fischer (1905)) and is particularly useful for the upper bounds. No analog of the new formulas has appeared before. The formulas not only enable us to recover or improve the main known results but also make a global change of the study on the topic. This will be illustrated by comparison of the new results with the known ones in geometry. Next, we will explain the mathematical tools for proving the results. That is, the trilogy of the recent development of the coupling theory: the Markovian coupling, the optimal Markovian coupling and the construction of distances for coupling. Finally, some related results and some problems for further study are also mentioned. It is hoped that the paper could be readable not only for probabilists but also for geometers and analysts.

On order-preservation and positive correlations for multidimensional diffusion processes

SummaryAs a continuation of the study by Herbst and Pitt (1991), this note presents two criteria. The first one is on the order-preservation for two (may be different) multidimensional diffusion processes. The second one is on the preservation of positive correlations for a diffusion process.

ON THREE CLASSICAL PROBLEMS FOR MARKOV CHAINS WITH CONTINUOUS TIME PARAMETERS

For a given transition rate, i.e., a Q-matrix Q = (qu) on a countable state space, the uniqueness of the Q-semigroup P(t) = (pu(t)), the recurrence and the positive recurrence of the corresponding Markov chain are three fundamental and classical problems, treated in many textbooks. As an addition, this paper introduces some practical results motivated from the study of a type of interacting particle systems, reaction diffusion processes. The main results are theorems (1. 11), (1.17) and (1.18). Their proofs are quite straightforward.

Nash inequalities for general symmetric forms

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 Poincaré and the logarithmic Sobolev inequalities in the present context.

Coupling, spectral gap and related topics (II)

THIS is the last one of a series of three papers. Here, we discuss six topics related to the spectral gap: the gradient estimate, the heat kernel and Harnack inequality, the logarithmic Sobolev inequality, the convergence in total variation, the algebraic convergence and the infinite-dimensional case. The perturbation of spectral gap and the logarithmic Sobolev constant under a linear transform is given (Theorem 5). A new proof for computing the logarithmic Sobolev constant in a basic case is also presented (Theorem 7).