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Author : Ariel Yadin
Publisher : Cambridge University Press
Page : 403 pages
File Size : 54,7 Mb
Release : 2024-05-31
Category : Mathematics
ISBN : 9781009123181
A modern introduction into the emerging research field of harmonic functions and random walks on groups.
Author : M. Picardello,W. Woess
Publisher : Cambridge University Press
Page : 378 pages
File Size : 49,7 Mb
Release : 1999-11-18
Category : Mathematics
ISBN : 0521773121
Comprehensive and interdisciplinary text covering the interplay between random walks and structure theory.
Author : Ariel Yadin
Publisher : Cambridge University Press
Page : 404 pages
File Size : 40,9 Mb
Release : 2024-05-31
Category : Mathematics
ISBN : 9781009546577
Research in recent years has highlighted the deep connections between the algebraic, geometric, and analytic structures of a discrete group. New methods and ideas have resulted in an exciting field, with many opportunities for new researchers. This book is an introduction to the area from a modern vantage point. It incorporates the main basics, such as Kesten's amenability criterion, Coulhon and Saloff-Coste inequality, random walk entropy and bounded harmonic functions, the Choquet–Deny Theorem, the Milnor–Wolf Theorem, and a complete proof of Gromov's Theorem on polynomial growth groups. The book is especially appropriate for young researchers, and those new to the field, accessible even to graduate students. An abundance of examples, exercises, and solutions encourage self-reflection and the internalization of the concepts introduced. The author also points to open problems and possibilities for further research.
Author : Steven P. Lalley
Publisher : Springer Nature
Page : 373 pages
File Size : 49,6 Mb
Release : 2023-05-08
Category : Mathematics
ISBN : 9783031256325
This text presents the basic theory of random walks on infinite, finitely generated groups, along with certain background material in measure-theoretic probability. The main objective is to show how structural features of a group, such as amenability/nonamenability, affect qualitative aspects of symmetric random walks on the group, such as transience/recurrence, speed, entropy, and existence or nonexistence of nonconstant, bounded harmonic functions. The book will be suitable as a textbook for beginning graduate-level courses or independent study by graduate students and advanced undergraduate students in mathematics with a solid grounding in measure theory and a basic familiarity with the elements of group theory. The first seven chapters could also be used as the basis for a short course covering the main results regarding transience/recurrence, decay of return probabilities, and speed. The book has been organized and written so as to be accessible not only to students in probability theory, but also to students whose primary interests are in geometry, ergodic theory, or geometric group theory.
Author : Wolfgang Woess
Publisher : Cambridge University Press
Page : 350 pages
File Size : 50,5 Mb
Release : 2000-02-13
Category : Mathematics
ISBN : 9780521552929
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 : B. Fiedler
Publisher : Gulf Professional Publishing
Page : 1099 pages
File Size : 44,7 Mb
Release : 2002-02-21
Category : Science
ISBN : 9780080532844
This handbook is volume II in a series collecting mathematical state-of-the-art surveys in the field of dynamical systems. Much of this field has developed from interactions with other areas of science, and this volume shows how concepts of dynamical systems further the understanding of mathematical issues that arise in applications. Although modeling issues are addressed, the central theme is the mathematically rigorous investigation of the resulting differential equations and their dynamic behavior. However, the authors and editors have made an effort to ensure readability on a non-technical level for mathematicians from other fields and for other scientists and engineers. The eighteen surveys collected here do not aspire to encyclopedic completeness, but present selected paradigms. The surveys are grouped into those emphasizing finite-dimensional methods, numerics, topological methods, and partial differential equations. Application areas include the dynamics of neural networks, fluid flows, nonlinear optics, and many others. While the survey articles can be read independently, they deeply share recurrent themes from dynamical systems. Attractors, bifurcations, center manifolds, dimension reduction, ergodicity, homoclinicity, hyperbolicity, invariant and inertial manifolds, normal forms, recurrence, shift dynamics, stability, to namejust a few, are ubiquitous dynamical concepts throughout the articles.
Author : B. Grigelionis,Yu. V. Prohorov,V. V. Sazonov,V. Statulevičius
Publisher : Walter de Gruyter GmbH & Co KG
Page : 624 pages
File Size : 53,6 Mb
Release : 2020-05-18
Category : Mathematics
ISBN : 9783112319024
No detailed description available for "PROB. TH. MATH. ST. ( GRIGELIONIS) VOL. 2 PROC.5/1989 E-BOOK".
Author : Tullio Ceccherini-Silberstein,Maura Salvatori,Ecaterina Sava-Huss
Publisher : Cambridge University Press
Page : 539 pages
File Size : 43,5 Mb
Release : 2017-06-29
Category : Mathematics
ISBN : 9781316604403
An up-to-date, panoramic account of the theory of random walks on groups and graphs, outlining connections with various mathematical fields.
Author : Daniel Lenz,Florian Sobieczky,Wolfgang Woess
Publisher : Springer Science & Business Media
Page : 345 pages
File Size : 53,7 Mb
Release : 2011-06-16
Category : Mathematics
ISBN : 9783034602440
These proceedings represent the current state of research on the topics 'boundary theory' and 'spectral and probability theory' of random walks on infinite graphs. They are the result of the two workshops held in Styria (Graz and St. Kathrein am Offenegg, Austria) between June 29th and July 5th, 2009. Many of the participants joined both meetings. Even though the perspectives range from very different fields of mathematics, they all contribute with important results to the same wonderful topic from structure theory, which, by extending a quotation of Laurent Saloff-Coste, could be described by 'exploration of groups by random processes'.
Author : Adam Korányi,Donald I. Cartwright
Publisher : American Mathematical Soc.
Page : 181 pages
File Size : 47,9 Mb
Release : 1997
Category : Mathematics
ISBN : 9780821806050
This volume presents the proceedings of the workshop 'Harmonic Functions on Graphs' held at the Graduate Center of CUNY in the fall of 1995. The main papers present material from four minicourses given by leading experts: D. Cartwright, A. Figa-Talamanca, S. Sawyer and T. Steger. These minicourses are introductions which gradually progress to deeper and less known branches of the subject. One of the topics treated is buildings, which are discrete analogues of symmetric spaces of arbitrary rank; buildings of rank are trees. Harmonic analysis on buildings is a fairly new and important field of research.One of the minicourses discusses buildings from the combinatorial perspective and another examines them from the $p$-adic perspective. The third minicourse deals with the connections of trees with $p$-adic analysis. And the fourth deals with random walks, i.e., with the probabilistic side of harmonic functions on trees. The book also contains the extended abstracts of 19 of the 20 lectures given by the participants on their recent results. These abstracts, well detailed and clearly understandable, give a good cross-section of the present state of research in the field.
Author : Vadim Kaimanovich
Publisher : Walter de Gruyter
Page : 545 pages
File Size : 41,8 Mb
Release : 2008-08-22
Category : Mathematics
ISBN : 9783110198089
Die jüngsten Entwicklungen zeigen, dass sich Wahrscheinlichkeitsverfahren zu einem sehr wirkungsvollen Werkzeug entwickelt haben, und das auf so unterschiedlichen Gebieten wie statistische Physik, dynamische Systeme, Riemann'sche Geometrie, Gruppentheorie, harmonische Analyse, Graphentheorie und Informatik.
Author : Ross G. Pinsky
Publisher : Cambridge University Press
Page : 492 pages
File Size : 49,7 Mb
Release : 1995-01-12
Category : Mathematics
ISBN : 9780521470148
In this book, Professor Pinsky gives a self-contained account of the theory of positive harmonic functions for second order elliptic operators, using an integrated probabilistic and analytic approach. The book begins with a treatment of the construction and basic properties of diffusion processes. This theory then serves as a vehicle for studying positive harmonic funtions. Starting with a rigorous treatment of the spectral theory of elliptic operators with nice coefficients on smooth, bounded domains, the author then develops the theory of the generalized principal eigenvalue, and the related criticality theory for elliptic operators on arbitrary domains. Martin boundary theory is considered, and the Martin boundary is explicitly calculated for several classes of operators. The book provides an array of criteria for determining whether a diffusion process is transient or recurrent. Also introduced are the theory of bounded harmonic functions, and Brownian motion on manifolds of negative curvature. Many results that form the folklore of the subject are here given a rigorous exposition, making this book a useful reference for the specialist, and an excellent guide for the graduate student.
Author : Frank Spitzer
Publisher : Springer Science & Business Media
Page : 419 pages
File Size : 55,9 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781475742299
This book is devoted exclusively to a very special class of random processes, namely, to random walk on the lattice points of ordinary Euclidian space. The author considers this high degree of specialization worthwhile because the theory of such random walks is far more complete than that of any larger class of Markov chains. Almost 100 pages of examples and problems are included.
Author : H. Heyer
Publisher : Springer Science & Business Media
Page : 491 pages
File Size : 54,8 Mb
Release : 2013-11-11
Category : Mathematics
ISBN : 9781489923646
The present volume contains the transactions of the lOth Oberwolfach Conference on "Probability Measures on Groups". The series of these meetings inaugurated in 1970 by L. Schmetterer and the editor is devoted to an intensive exchange of ideas on a subject which developed from the relations between various topics of mathematics: measure theory, probability theory, group theory, harmonic analysis, special functions, partial differential operators, quantum stochastics, just to name the most significant ones. Over the years the fruitful interplay broadened in various directions: new group-related structures such as convolution algebras, generalized translation spaces, hypercomplex systems, and hypergroups arose from generalizations as well as from applications, and a gradual refinement of the combinatorial, Banach-algebraic and Fourier analytic methods led to more precise insights into the theory. In a period of highest specialization in scientific thought the separated minds should be reunited by actively emphasizing similarities, analogies and coincidences between ideas in their fields of research. Although there is no real separation between one field and another - David Hilbert denied even the existence of any difference between pure and applied mathematics - bridges between probability theory on one side and algebra, topology and geometry on the other side remain absolutely necessary. They provide a favorable ground for the communication between apparently disjoint research groups and motivate the framework of what is nowadays called "Structural probability theory".
Author : Yves Benoist,Jean-François Quint
Publisher : Springer
Page : 323 pages
File Size : 41,9 Mb
Release : 2016-10-20
Category : Mathematics
ISBN : 9783319477213
The classical theory of random walks describes the asymptotic behavior of sums of independent identically distributed random real variables. This book explains the generalization of this theory to products of independent identically distributed random matrices with real coefficients. Under the assumption that the action of the matrices is semisimple – or, equivalently, that the Zariski closure of the group generated by these matrices is reductive - and under suitable moment assumptions, it is shown that the norm of the products of such random matrices satisfies a number of classical probabilistic laws. This book includes necessary background on the theory of reductive algebraic groups, probability theory and operator theory, thereby providing a modern introduction to the topic.
