Description : The first and only book to make this research available in the West Concise and accessible: proofs and other technical matters are kept to a minimum to help the non-specialist Each chapter is self-contained to make the book easy-to-use
Description : In this book, the functional inequalities are introduced to describe: (i) the spectrum of the generator: the essential and discrete spectrums, high order eigenvalues, the principle eigenvalue, and the spectral gap; (ii) the semigroup properties: the uniform intergrability, the compactness, the convergence rate, and the existence of density; (iii) the reference measure and the intrinsic metric: the concentration, the isoperimetic inequality, and the transportation cost inequality.
Description : ' This book is representative of the work of Chinese probabilists on probability theory and its applications in physics. It presents a unique treatment of general Markov jump processes: uniqueness, various types of ergodicity, Markovian couplings, reversibility, spectral gap, etc. It also deals with a typical class of non-equilibrium particle systems, including the typical Schlögl model taken from statistical physics. The constructions, ergodicity and phase transitions for this class of Markov interacting particle systems, namely, reaction–diffusion processes, are presented. In this new edition, a large part of the text has been updated and two-and-a-half chapters have been rewritten. The book is self-contained and can be used in a course on stochastic processes for graduate students. Contents:An Overview of the Book: Starting from Markov ChainsGeneral Jump Processes:Transition Function and Its Laplace TransformExistence and Simple Constructions of Jump ProcessesUniqueness CriteriaRecurrence, Ergodicity and Invariant MeasuresProbability Metrics and Coupling MethodsSymmetrizable Jump Processes:Symmetrizable Jump Processes and Dirichlet FormsField TheoryLarge DeviationsSpectral GapEquilibrium Particle Systems:Random FieldsReversible Spin Processes and Exclusion ProcessesYang–Mills Lattice FieldNon-Equilibrium Particle Systems:Constructions of the ProcessesExistence of Stationary Distributions and ErgodicityPhase TransitionsHydrodynamic Limits Readership: Upper-level undergraduates, graduate students and researchers in mathematical physics, probability and statistics. Keywords:Jump Process;Reversible Jump Process;Spectral Gap;Coupling Method;Large Deviation;Random Field;Reaction-Diffusion Process;Hydrodynamic LimitReviews:“… this book should be very useful to anyone interested in Markov processes, because it can serve as an introduction to a large variety of subjects and models, as well as an account of some recent works of Chinese probabilists.”Mathematical Reviews Reviews of the First Edition: “More recently, the school led by the author in Beijing has worked on the construction and ergodic theory of the class of interacting particle systems known as reaction diffusion processes … This book provides a useful account of a substantial portion of the work of Chinese probabilists over the past two decades, much of which has been relatively inaccessible to Western workers … this is a useful reference work for probabilists working in these areas, and a contribution to international communication in probability theory.”Mathematical Reviews “The book is a comprehensive account of the theory of jump processes and particle systems. The author is an outstanding Chinese specialist in probability theory and stochastic processes creating the Chinese school of Markov processes.”Zentralblatt fur Mathematik “He did a lot to popularize the subject in China and with Yan Shi-Jian was instrumental in having the second special year 1988–89 at the Nankai Institute devoted to probability … The book is admirable not only for the circumstances in which it is written but for what it contains … Chen's book contains a wide variety of topics that cannot be found in any other place … if you are curious about interacting particle systems this book belongs on your bookshelf.”SIAM Reviews '
Description : This volume is a collection of 15 research and survey papers written by the speakers from two international conferences held in Japan, The 11th Mathematical Society of Japan International Research Institute's Stochastic Analysis on Large Scale Interacting Systems and Stochastic Analysis and Statistical Mechanics. Topics discussed in the volume cover the hydrodynamic limit, fluctuations, large deviations, spectral gap (Poincare inequality), logarithmic Sobolev inequality, Ornstein-Zernike asymptotics, random environments, determinantal expressions for systems including exclusion processes (stochastic lattice gas, Kawasaki dynamics), zero range processes, interacting Brownian particles, random walks, self-avoiding walks, Ginzburg-Landau model, interface models, Ising model, Widom-Rowlinson model, directed polymers, random matrices, Dyson's model, and more. The material is suitable for graduate students and researchers interested in probability theory, stochastic processes, and statistical mechanics.
Description : Inequalities play an important role in almost all branches of mathematics as well as in other areas of science and engineering. This book surveys the present state of the theory of weighted integral inequalities of Hardy type, including modifications concerning Hardy-Steklov operators, and some basic results about Hardy type inequalities and their limit (Carleman-Knopp type) inequalities. It also describes some rather new fields such as higher order and fractional order Hardy type inequalities and integral inequalities on the cone of monotone functions together with some applications and open problems. The book can serve as a reference and a source of inspiration for researchers working in these and related areas, but could also be used for advanced graduate courses.
Description : Concerned with probability theory, Elton Hsu's study focuses primarily on the relations between Brownian motion on a manifold and analytical aspects of differential geometry. A key theme is the probabilistic interpretation of the curvature of a manifold.