Branching Random Walks
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| Author | : Zhan Shi |
| Publisher | : Springer |
| Total Pages | : 143 |
| Release | : 2016-02-04 |
| Genre | : Mathematics |
| ISBN | : 3319253727 |
Providing an elementary introduction to branching random walks, the main focus of these lecture notes is on the asymptotic properties of one-dimensional discrete-time supercritical branching random walks, and in particular, on extreme positions in each generation, as well as the evolution of these positions over time. Starting with the simple case of Galton-Watson trees, the text primarily concentrates on exploiting, in various contexts, the spinal structure of branching random walks. The notes end with some applications to biased random walks on trees.
| Author | : Elena Yarovaya |
| Publisher | : John Wiley & Sons |
| Total Pages | : 0 |
| Release | : 2023-06-14 |
| Genre | : Science |
| ISBN | : 9781848212084 |
The book is devoted to a modern section of the probability theory, the so-called theory of branching random walks. Chapter 1 describes the random walk model in the finite branching one-source environment. Chapter 2 is devoted to a model of homogeneous, symmetrical, irreducible random walk (without branching) with finite variance of the jumps on the multidimensional integer continuous-time lattice where transition is possible to an arbitrary point of the lattice and not only to the neighbor state. This model is a generalization of the simple symmetrical random walk often encountered in the applied studies. In Chapter 3 the branching random walk is studied by means of the spectral methods. Here, the property of monotonicity of the mean number of particles in the source plays an important role in the subsequent parts of the book. Chapter 4 demonstrates that existence of an isolated positive eigenvalue in the spectrum of unperturbed random walk generator defines the exponential growth of the process in the supercritical case. Chapter 5 exemplify application of the Tauberian theorems in the asymptotical problems of the probability theory. At last, the final Chapters 6 and 7 are devoted to detailed examination of survival probabilities in the critical and subcritical cases.
| Author | : Gregory F. Lawler |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 226 |
| Release | : 2012-11-06 |
| Genre | : Mathematics |
| ISBN | : 1461459729 |
A central study in Probability Theory is the behavior of fluctuation phenomena of partial sums of different types of random variable. One of the most useful concepts for this purpose is that of the random walk which has applications in many areas, particularly in statistical physics and statistical chemistry. Originally published in 1991, Intersections of Random Walks focuses on and explores a number of problems dealing primarily with the nonintersection of random walks and the self-avoiding walk. Many of these problems arise in studying statistical physics and other critical phenomena. Topics include: discrete harmonic measure, including an introduction to diffusion limited aggregation (DLA); the probability that independent random walks do not intersect; and properties of walks without self-intersections. The present softcover reprint includes corrections and addenda from the 1996 printing, and makes this classic monograph available to a wider audience. With a self-contained introduction to the properties of simple random walks, and an emphasis on rigorous results, the book will be useful to researchers in probability and statistical physics and to graduate students interested in basic properties of random walks.
| Author | : Ming Fang |
| Publisher | : |
| Total Pages | : 81 |
| Release | : 2011 |
| Genre | : |
| ISBN | : |
| Author | : Vladimir V. Rykov |
| Publisher | : Springer |
| Total Pages | : 551 |
| Release | : 2017-12-21 |
| Genre | : Computers |
| ISBN | : 3319715046 |
This book constitutes the refereed proceedings of the First International Conference on Analytical and Computational Methods in Probability Theory and its Applications, ACMPT 2017, held in Moscow, Russia, in October 2017. The 42 full papers presented were carefully reviewed and selected from 173 submissions. The conference program consisted of four main themes associated with significant contributions made by A.D.Soloviev. These are: Analytical methods in probability theory, Computational methods in probability theory, Asymptotical methods in probability theory, the history of mathematics.
| Author | : Russell Lyons |
| Publisher | : Cambridge University Press |
| Total Pages | : 1023 |
| Release | : 2017-01-20 |
| Genre | : Mathematics |
| ISBN | : 1316785335 |
Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.
| Author | : Jim Pitman |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 257 |
| Release | : 2006-05-11 |
| Genre | : Mathematics |
| ISBN | : 354030990X |
The purpose of this text is to bring graduate students specializing in probability theory to current research topics at the interface of combinatorics and stochastic processes. There is particular focus on the theory of random combinatorial structures such as partitions, permutations, trees, forests, and mappings, and connections between the asymptotic theory of enumeration of such structures and the theory of stochastic processes like Brownian motion and Poisson processes.
| Author | : Steven P. Lalley |
| Publisher | : Springer Nature |
| Total Pages | : 373 |
| Release | : 2023-05-08 |
| Genre | : Mathematics |
| ISBN | : 3031256328 |
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 | : J. T. Cox |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 114 |
| Release | : 2004 |
| Genre | : Mathematics |
| ISBN | : 0821835424 |
Studies the evolution of the large finite spatial systems in size-dependent time scales and compare them with the behavior of the infinite systems, which amounts to establishing the so-called finite system scheme. This title introduces the concept of a continuum limit in the hierarchical mean field limit.
| Author | : A. A. Borovkov |
| Publisher | : Cambridge University Press |
| Total Pages | : 437 |
| Release | : 2020-10-29 |
| Genre | : Mathematics |
| ISBN | : 1108901204 |
This is a companion book to Asymptotic Analysis of Random Walks: Heavy-Tailed Distributions by A.A. Borovkov and K.A. Borovkov. Its self-contained systematic exposition provides a highly useful resource for academic researchers and professionals interested in applications of probability in statistics, ruin theory, and queuing theory. The large deviation principle for random walks was first established by the author in 1967, under the restrictive condition that the distribution tails decay faster than exponentially. (A close assertion was proved by S.R.S. Varadhan in 1966, but only in a rather special case.) Since then, the principle has always been treated in the literature only under this condition. Recently, the author jointly with A.A. Mogul'skii removed this restriction, finding a natural metric for which the large deviation principle for random walks holds without any conditions. This new version is presented in the book, as well as a new approach to studying large deviations in boundary crossing problems. Many results presented in the book, obtained by the author himself or jointly with co-authors, are appearing in a monograph for the first time.
