Ebook: Fluctuations in Markov Processes: Time Symmetry and Martingale Approximation

The present volume contains the most advanced theories on the martingale approach to central limit theorems. Using the time symmetry properties of the Markov processes, the book develops the techniques that allow us to deal with infinite dimensional models that appear in statistical mechanics and engineering (interacting particle systems, homogenization in random environments, and diffusion in turbulent flows, to mention just a few applications). The first part contains a detailed exposition of the method, and can be used as a text for graduate courses. The second concerns application to exclusion processes, in which the duality methods are fully exploited. The third part is about the homogenization of diffusions in random fields, including passive tracers in turbulent flows (including the superdiffusive behavior).

There are no other books in the mathematical literature that deal with this kind of approach to the problem of the central limit theorem. Hence, this volume meets the demand for a monograph on this powerful approach, now widely used in many areas of probability and mathematical physics. The book also covers the connections with and application to hydrodynamic limits and homogenization theory, so besides probability researchers it will also be of interest also to mathematical physicists and analysts.

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“To design future networks that are worthy of society’s trust, we must put the ‘discipline’ of computer networking on a much stronger foundation. This book rises above the considerable minutiae of today’s networking technologies to emphasize the long-standing mathematical underpinnings of the field.” –Professor Jennifer Rexford, Department of Computer Science, Princeton University   “This book is exactly the one I have been waiting for the last couple of years. Recently, I decided most students were already very familiar with the way the net works but were not being taught the fundamentals–the math. This book contains the knowledge for people who will create and understand future communications systems." –Professor Jon Crowcroft, The Computer Laboratory, University of Cambridge   The Essential Mathematical Principles Required to Design, Implement, or Evaluate Advanced Computer Networks   Students, researchers, and professionals in computer networking require a firm conceptual understanding of its foundations. Mathematical Foundations of Computer Networking provides an intuitive yet rigorous introduction to these essential mathematical principles and techniques.   Assuming a basic grasp of calculus, this book offers sufficient detail to serve as the only reference many readers will need. Each concept is described in four ways: intuitively; using appropriate mathematical notation; with a numerical example carefully chosen for its relevance to networking; and with a numerical exercise for the reader.   The first part of the text presents basic concepts, and the second part introduces four theories in a progression that has been designed to gradually deepen readers’ understanding. Within each part, chapters are as self-contained as possible.   The first part covers probability; statistics; linear algebra; optimization; and signals, systems, and transforms. Topics range from Bayesian networks to hypothesis testing, and eigenvalue computation to Fourier transforms.   These preliminary chapters establish a basis for the four theories covered in the second part of the book: queueing theory, game theory, control theory, and information theory. The second part also demonstrates how mathematical concepts can be applied to issues such as contention for limited resources, and the optimization of network responsiveness, stability, and throughput Part 1. General Theory -- A Warming-Up Example -- Central Limit Theorems -- Random Walks in Random Environment -- Bounds and Variational Principles for the Asymptotic Variance -- Part 2. Simple Exclusion Processes -- The Simple Exclusion Process -- Self-diffusion -- Equilibrium Fluctuations of the Density Field -- Regularity of the Asymptotic Variance -- Part 3. Diffusions in Random Environments -- Diffusions in Random Environments -- Variational Principles for the Limiting Variance -- Diffusions with Divergence Free Drifts -- Diffusions with Gaussian Drifts -- Ornstein-Uhlenbeck Process with a Random Potential -- Analytic Methods in Homogenization Theory