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Random Walks And Heat Kernels On Graphs
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Book Synopsis Random Walks and Heat Kernels on Graphs by : Martin T. Barlow
Download or read book Random Walks and Heat Kernels on Graphs written by Martin T. Barlow and published by Cambridge University Press. This book was released on 2017-02-23 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: This introduction to random walks on infinite graphs gives particular emphasis to graphs with polynomial volume growth. It offers an overview of analytic methods, starting with the connection between random walks and electrical resistance, and then proceeding to study the use of isoperimetric and Poincaré inequalities. The book presents rough isometries and looks at the properties of a graph that are stable under these transformations. Applications include the 'type problem': determining whether a graph is transient or recurrent. The final chapters show how geometric properties of the graph can be used to establish heat kernel bounds, that is, bounds on the transition probabilities of the random walk, and it is proved that Gaussian bounds hold for graphs that are roughly isometric to Euclidean space. Aimed at graduate students in mathematics, the book is also useful for researchers as a reference for results that are hard to find elsewhere.
Book Synopsis Random Walks and Heat Kernels on Graphs by : M. T. Barlow
Download or read book Random Walks and Heat Kernels on Graphs written by M. T. Barlow and published by Cambridge University Press. This book was released on 2017-02-23 with total page 239 pages. Available in PDF, EPUB and Kindle. Book excerpt: Useful but hard-to-find results enrich this introduction to the analytic study of random walks on infinite graphs.
Book Synopsis Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces by : Pascal Auscher
Download or read book Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces written by Pascal Auscher and published by American Mathematical Soc.. This book was released on 2003 with total page 434 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the expanded lecture notes of courses taught at the Emile Borel Centre of the Henri Poincare Institute (Paris). In the book, leading experts introduce recent research in their fields. The unifying theme is the study of heat kernels in various situations using related geometric and analytic tools. Topics include analysis of complex-coefficient elliptic operators, diffusions on fractals and on infinite-dimensional groups, heat kernel and isoperimetry on Riemannian manifolds, heat kernels and infinite dimensional analysis, diffusions and Sobolev-type spaces on metric spaces, quasi-regular mappings and $p$-Laplace operators, heat kernel and spherical inversion on $SL 2(C)$, random walks and spectral geometry on crystal lattices, isoperimetric and isocapacitary inequalities, and generating function techniques for random walks on graphs. This volume is suitable for graduate students and research mathematicians interested in random processes and analysis on manifolds.
Book Synopsis Random Walks on Disordered Media and their Scaling Limits by : Takashi Kumagai
Download or read book Random Walks on Disordered Media and their Scaling Limits written by Takashi Kumagai and published by Springer. This book was released on 2014-01-25 with total page 147 pages. Available in PDF, EPUB and Kindle. Book excerpt: In these lecture notes, we will analyze the behavior of random walk on disordered media by means of both probabilistic and analytic methods, and will study the scaling limits. We will focus on the discrete potential theory and how the theory is effectively used in the analysis of disordered media. The first few chapters of the notes can be used as an introduction to discrete potential theory. Recently, there has been significant progress on the theory of random walk on disordered media such as fractals and random media. Random walk on a percolation cluster(‘the ant in the labyrinth’)is one of the typical examples. In 1986, H. Kesten showed the anomalous behavior of a random walk on a percolation cluster at critical probability. Partly motivated by this work, analysis and diffusion processes on fractals have been developed since the late eighties. As a result, various new methods have been produced to estimate heat kernels on disordered media. These developments are summarized in the notes.
Book Synopsis Random Walk and the Heat Equation by : Gregory F. Lawler
Download or read book Random Walk and the Heat Equation written by Gregory F. Lawler and published by American Mathematical Soc.. This book was released on 2010-11-22 with total page 170 pages. Available in PDF, EPUB and Kindle. Book excerpt: The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation and considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equations and the closely related notion of harmonic functions from a probabilistic perspective. The theme of the first two chapters of the book is the relationship between random walks and the heat equation. This first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalization of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set. The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.
Book Synopsis Spectral Graph Theory by : Fan R. K. Chung
Download or read book Spectral Graph Theory written by Fan R. K. Chung and published by American Mathematical Soc.. This book was released on 1997 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text discusses spectral graph theory.
Book Synopsis Heat Kernel and Analysis on Manifolds by : Alexander Grigoryan
Download or read book Heat Kernel and Analysis on Manifolds written by Alexander Grigoryan and published by American Mathematical Soc.. This book was released on 2009 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: "This volume contains the expanded lecture notes of courses taught at the Emile Borel Centre of the Henri Poincaré Institute (Paris). In the book, leading experts introduce recent research in their fields. The unifying theme is the study of heat kernels in various situations using related geometric and analytic tools. Topics include analysis of complex-coefficient elliptic operators, diffusions on fractals and on infinite-dimensional groups, heat kernel and isoperimetry on Riemannian manifolds, heat kernels and infinite dimensional analysis, diffusions and Sobolev-type spaces on metric spaces, quasi-regular mappings and p -Laplace operators, heat kernel and spherical inversion on SL 2 (C) , random walks and spectral geometry on crystal lattices, isoperimetric and isocapacitary inequalities, and generating function techniques for random walks on graphs."--Publisher's website.
Book Synopsis The Art of Random Walks by : Andras Telcs
Download or read book The Art of Random Walks written by Andras Telcs and published by Springer. This book was released on 2006-10-18 with total page 200 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main aim of this book is to reveal connections between the physical and geometric properties of space and diffusion. This is done in the context of random walks in the absence of algebraic structure, local or global spatial symmetry or self-similarity. The author studies heat diffusion at this general level and discusses the multiplicative Einstein relation; Isoperimetric inequalities; and Heat kernel estimates; Elliptic and parabolic Harnack inequality.
Book Synopsis Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot by : Michel Laurent Lapidus
Download or read book Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot written by Michel Laurent Lapidus and published by American Mathematical Soc.. This book was released on 2004 with total page 592 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume offers an excellent selection of cutting-edge articles about fractal geometry, covering the great breadth of mathematics and related areas touched by this subject. Included are rich survey articles and fine expository papers. The high-quality contributions to the volume by well-known researchers--including two articles by Mandelbrot--provide a solid cross-section of recent research representing the richness and variety of contemporary advances in and around fractal geometry. In demonstrating the vitality and diversity of the field, this book will motivate further investigation into the many open problems and inspire future research directions. It is suitable for graduate students and researchers interested in fractal geometry and its applications. This is a two-part volume. Part 1 covers analysis, number theory, and dynamical systems; Part 2, multifractals, probability and statistical mechanics, and applications.
Book Synopsis Introduction to Analysis on Graphs by : Alexander Grigor’yan
Download or read book Introduction to Analysis on Graphs written by Alexander Grigor’yan and published by American Mathematical Soc.. This book was released on 2018-08-23 with total page 150 pages. Available in PDF, EPUB and Kindle. Book excerpt: A central object of this book is the discrete Laplace operator on finite and infinite graphs. The eigenvalues of the discrete Laplace operator have long been used in graph theory as a convenient tool for understanding the structure of complex graphs. They can also be used in order to estimate the rate of convergence to equilibrium of a random walk (Markov chain) on finite graphs. For infinite graphs, a study of the heat kernel allows to solve the type problem—a problem of deciding whether the random walk is recurrent or transient. This book starts with elementary properties of the eigenvalues on finite graphs, continues with their estimates and applications, and concludes with heat kernel estimates on infinite graphs and their application to the type problem. The book is suitable for beginners in the subject and accessible to undergraduate and graduate students with a background in linear algebra I and analysis I. It is based on a lecture course taught by the author and includes a wide variety of exercises. The book will help the reader to reach a level of understanding sufficient to start pursuing research in this exciting area.
Book Synopsis Random Walks on Infinite Graphs and Groups by : Wolfgang Woess
Download or read book Random Walks on Infinite Graphs and Groups written by Wolfgang Woess and published by Cambridge University Press. This book was released on 2000-02-13 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: The main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. The author considers Markov chains whose state space is equipped with the structure of an infinite, locally finite graph, or as a particular case, of a finitely generated group. The transition probabilities are assumed to be adapted to the underlying structure in some way that must be specified precisely in each case. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk. Vice-versa, random walks may also be seen as useful tools for classifying, or at least describing the structure of graphs and groups. Links with spectral theory and discrete potential theory are also discussed. This book will be essential reading for all researchers working in stochastic process and related topics.
Author :Tullio Ceccherini-Silberstein Publisher :Cambridge University Press ISBN 13 :1316604403 Total Pages :539 pages Book Rating :4.3/5 (166 download)
Book Synopsis Groups, Graphs and Random Walks by : Tullio Ceccherini-Silberstein
Download or read book Groups, Graphs and Random Walks written by Tullio Ceccherini-Silberstein and published by Cambridge University Press. This book was released on 2017-06-29 with total page 539 pages. Available in PDF, EPUB and Kindle. Book excerpt: An up-to-date, panoramic account of the theory of random walks on groups and graphs, outlining connections with various mathematical fields.
Book Synopsis Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs by : Alexander Grigor'yan
Download or read book Analysis and Partial Differential Equations on Manifolds, Fractals and Graphs written by Alexander Grigor'yan and published by Walter de Gruyter GmbH & Co KG. This book was released on 2021-01-18 with total page 337 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book covers the latest research in the areas of mathematics that deal the properties of partial differential equations and stochastic processes on spaces in connection with the geometry of the underlying space. Written by experts in the field, this book is a valuable tool for the advanced mathematician.
Book Synopsis Markov Chains and Mixing Times: Second Edition by : David A. Levin
Download or read book Markov Chains and Mixing Times: Second Edition written by David A. Levin and published by American Mathematical Soc.. This book was released on 2017-10-31 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is an introduction to the modern theory of Markov chains, whose goal is to determine the rate of convergence to the stationary distribution, as a function of state space size and geometry. This topic has important connections to combinatorics, statistical physics, and theoretical computer science. Many of the techniques presented originate in these disciplines. The central tools for estimating convergence times, including coupling, strong stationary times, and spectral methods, are developed. The authors discuss many examples, including card shuffling and the Ising model, from statistical mechanics, and present the connection of random walks to electrical networks and apply it to estimate hitting and cover times. The first edition has been used in courses in mathematics and computer science departments of numerous universities. The second edition features three new chapters (on monotone chains, the exclusion process, and stationary times) and also includes smaller additions and corrections throughout. Updated notes at the end of each chapter inform the reader of recent research developments.
Book Synopsis Stochastic Processes: Theory and Methods by : D N Shanbhag
Download or read book Stochastic Processes: Theory and Methods written by D N Shanbhag and published by Gulf Professional Publishing. This book was released on 2001 with total page 990 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume in the series contains chapters on areas such as pareto processes, branching processes, inference in stochastic processes, Poisson approximation, Levy processes, and iterated random maps and some classes of Markov processes. Other chapters cover random walk and fluctuation theory, a semigroup representation and asymptomatic behavior of certain statistics of the Fisher-Wright-Moran coalescent, continuous-time ARMA processes, record sequence and their applications, stochastic networks with product form equilibrium, and stochastic processes in insurance and finance. Other subjects include renewal theory, stochastic processes in reliability, supports of stochastic processes of multiplicity one, Markov chains, diffusion processes, and Ito's stochastic calculus and its applications. c. Book News Inc.
Book Synopsis Random Walks and Electric Networks by : Peter G. Doyle
Download or read book Random Walks and Electric Networks written by Peter G. Doyle and published by American Mathematical Soc.. This book was released on 1984-12-31 with total page 159 pages. Available in PDF, EPUB and Kindle. Book excerpt: Probability theory, like much of mathematics, is indebted to physics as a source of problems and intuition for solving these problems. Unfortunately, the level of abstraction of current mathematics often makes it difficult for anyone but an expert to appreciate this fact. Random Walks and electric networks looks at the interplay of physics and mathematics in terms of an example—the relation between elementary electric network theory and random walks —where the mathematics involved is at the college level.
Author :Tullio Ceccherini-Silberstein Publisher :Cambridge University Press ISBN 13 :1316817784 Total Pages :539 pages Book Rating :4.3/5 (168 download)
Book Synopsis Groups, Graphs and Random Walks by : Tullio Ceccherini-Silberstein
Download or read book Groups, Graphs and Random Walks written by Tullio Ceccherini-Silberstein and published by Cambridge University Press. This book was released on 2017-06-29 with total page 539 pages. Available in PDF, EPUB and Kindle. Book excerpt: An accessible and panoramic account of the theory of random walks on groups and graphs, stressing the strong connections of the theory with other branches of mathematics, including geometric and combinatorial group theory, potential analysis, and theoretical computer science. This volume brings together original surveys and research-expository papers from renowned and leading experts, many of whom spoke at the workshop 'Groups, Graphs and Random Walks' celebrating the sixtieth birthday of Wolfgang Woess in Cortona, Italy. Topics include: growth and amenability of groups; Schrödinger operators and symbolic dynamics; ergodic theorems; Thompson's group F; Poisson boundaries; probability theory on buildings and groups of Lie type; structure trees for edge cuts in networks; and mathematical crystallography. In what is currently a fast-growing area of mathematics, this book provides an up-to-date and valuable reference for both researchers and graduate students, from which future research activities will undoubtedly stem.
