Lyapunov conditions for Super Poincaré inequalities

We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincaré inequality (for instance logarithmic Sobolev or F-Sobolev). The case of Poincaré and weak Poincaré inequalities was studied in [D. Bakry, P. Cattiaux, A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal. 254 (3) (2008) 727-759. Available on Mathematics arXiv:math.PR/0703355, 2007]. This approach allows us to recover and extend in a unified way some known criteria in the euclidean case (Bakry and Emery, Wang, Kusuoka and Stroock, …).     

Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré

We study the relationship between two classical approaches for quantitative ergodic properties: the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of Poincaré type). We show that they can be linked through new inequalities (Lyapunov-Poincaré inequalities). Explicit examples for diffusion processes are studied, improving some results in the literature. The example of the kinetic Fokker-Planck equation recently studied by Hérau and Nier, Helffer and Nier, and Villani is in particular discussed in the final section.     

On the best constant of weighted Poincaré inequalities

We compute the optimal constant for some weighted Poincaré inequalities obtained by Fausto Ferrari and Enrico Valdinoci in [F. Ferrari, E. Valdinoci, Some weighted Poincaré inequalities, Indiana Univ. Math. J. 58 (4) (2009) 1619-1637].     

On the Empty Essential Spectrum for Markov Processes in Dimension One

This paper gives characterizations for diffusion processes on the line and birth-death processes whose generators admit the empty essential spectra. Some equivalent conditions for empty essential spectra for general Markov generators are also discussed.     

一维Markov过程的Nash不等式

本文得到了生灭过程和一维扩散过程满足Nash不等式的判别准则,并证明 了对此二类过程,非常返性蕴含相应半群如下收敛速度||P(t)||1→∞≤Ct-1.同时也给 出一般马氏链满足Nash不等式的充分条件.     

Symmetrization of Lévy processes and applications

It is shown that many of the classical generalized isoperimetric inequalities for the Laplacian when viewed in terms of Brownian motion extend to a wide class of Lévy processes. The results are derived from the multiple integral inequalities of Brascamp, Lieb and Luttinger but the probabilistic structure of the processes plays a crucial role in the proofs.     

Capacitary Criteria for Poincaré-Type Inequalities

The Poincaré-type inequality is a unification of various inequalities including the F-Sobolev inequalities, Sobolev-type inequalities, logarithmic Sobolev inequalities, and so on. The aim of this paper is to deduce some unified upper and lower bounds of the optimal constants in Poincaré-type inequalities for a large class of normed linear (Banach, Orlicz) spaces in terms of capacity. The lower and upper bounds differ only by a multiplicative constant, and so the capacitary criteria for the inequalities are also established. Both the transient and the ergodic cases are treated. Besides, the explicit lower and upper estimates in dimension one are computed. Mathematics Subject Classifications (2000) 60J55, 31C25, 60J35, 47D07.Research supported in part NSFC (No. 10121101) and 973 Project.     

Harnack inequalities for stochastic equations driven by Lévy noise

By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semilinear stochastic equations in Hilbert spaces.     

Local Uniqueness of Solutions of General Variational Inequalities

We use the concept of Fréchet approximate Jacobian matrices to establish the local uniqueness of solutions to general variational inequalities which involve continuous, not necessarily locally Lipschitz continuous data.     

Large Deviations of U-Statistics. I

We develop large-deviation results with explicit order terms and Cramér's series for nondegenerate U-statistics of degree m under Cramér-type conditions on the kernel. The method of the proof is based on the contraction technique of Keener, Robinson, and Weber [15], which is a natural generalization of the classical method of Cramér [10]. Other techniques used in the proofs include truncation, decoupling inequalities, Borell's inequality for Rademacher chaos, and a partitioning method to bound the degenerate remainder term.     

Nash Inequalities for Markov Processes in Dimension One

In this paper, we give characterizations of Nash inequalities for birth-death process and diffusion process on the line. As a by-product, we prove that for these processes, transience implies that the semigroups P(t) decay as ∥P(t)∥_1→∞≥Ct ₋₁. Sufficient conditions for general Markov chains are also obtained. Received November 29, 2000, Revised February 21, 2001, Accepted March 16, 2001     

Variational Formulas of Poincaré-type Inequalities for Birth-Death Processes

Abstract In author’s one previous paper, the same topic was studied for one dimensional diffusions. As a continuation, this paper studies the discrete case, that is the birth-death processes. The explicit criteria for the inequalities, the variational formulas and explicit bounds of the corresponding constants in the inequalities are presented. As typical applications, the Nash inequalities and logarithmic Sobolev inequalities are examined. Research supported in part by NSFC (No. 10121101), 973 Project and RFDP     

A Berry-Esséen Theorem for Serial Rank Statistics

Berry-Esséen bounds of the optimal O(n-1/2) order are obtained, under the null hypothesis of randomness, for serial linear rank statistics, of the form a1 (Rt)a2(Rt-k). Such statistics play an essential role in distribution-free methods for time-series analysis, where they provide nonparametric analogues to classical (Gaussian) correlogram-based methods. Berry-Esséen inequalities are established under mild conditions on the score-generating functions, allowing for normal (van der Waerden) scores. They extend to the serial case the earlier result of Does (1982, Ann. Probab., 10, 982-991) on (nonserial) linear rank statistics, and to the context of nonparametric rank-based statistics the parametric results of Taniguchi (1991, Higher Order Asymptotics for Time Series Analysis, Springer, New York) on quadratic forms of Gaussian stationary processes.     

Is an Orlicz–Poincaré inequality an open ended condition,and what does that mean?

In [16], Keith and Zhong prove that spaces admitting Poincaré inequalities also admit a priori stronger Poincaré inequalities. We use their technique, with slight adjustments, to obtain a similar result in the case of Orlicz–Poincaré inequalities. We give examples in the plane that show all hypotheses are required.     

Self-improving properties of inequalities of Poincaré type on measure spaces and applications

We show that the self-improving nature of Poincaré estimates persists for domains in rather general measure spaces. We consider both weak type and strong type inequalities, extending techniques of B. Franchi, C. Pérez and R. Wheeden. As an application in spaces of homogeneous type, we derive global Poincaré estimates for a class of domains with rough boundaries that we call ?-John domains, and we show that such domains have the requisite properties. This class includes John (or Boman) domains as well as s-John domains. Further applications appear in a companion paper.     

Mass transportation proofs of free functional inequalities, and free Poincaré inequalities

This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincaré inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on R₊ with the reference example of the Marcenko-Pastur distribution.     

Refined convex Sobolev inequalities

This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy-entropy production relation improves the known inequalities of this type. The corresponding generalized Poincaré-type inequalities with weights are derived. Optimal constants are compared to the usual Poincaré constant.     

Criteria for Feller transition functions

This paper presents some conditions for the minimal Q-function to be a Feller transition function, for a given q-matrix Q. We derive a sufficient condition that is stated explicitly in terms of the transition rates. Furthermore, some necessary and sufficient conditions are derived of a more implicit nature, namely in terms of properties of a system of equations (or inequalities) and in terms of the operator induced by the q-matrix. The criteria lead to some perturbation results. These results are applied to birth-death processes with killing, yielding some sufficient and some necessary conditions for the Feller property directly in terms of the rates. An essential step in the analysis is the idea of associating the Feller property with individual states.     

Monotone matrices and monotone Markov processes

A stochastic matrix is “monotone” [4] if its row-vectors are stochastically increasing. Closure properties, characterizations and the availability of a second maximal eigenvalue are developed. Such monotonicity is present in a variety of processes in discrete and continous time. In particular, birth-death processes are monotone. Conditions for the sequential monotonicity of a process are given and related inequalities presented.     

Weak log-Sobolev and L p weak Poincaré inequalities for general symmetric forms

Weak log-Sobolev and L^p weak Poincaré inequalities for general symmetric forms are investigated by using newly defined Cheeger’s isoperimetric constants. Some concrete examples of ergodic birth-death processes are also presented to illustrate the results.