Some Case Studies Of Random Walks In Dynamic Random Environments

The simplest mathematical model of the Brownian motion of physics is the simple, symmetric random walk. This book collects and compares current results OCo mostly strong theorems which describe the properties of a random walk. The modern problems of the limit theorems of probability theory are treated in the simple case of coin tossing. Taking advantage of this simplicity, the reader is familiarized with limit theorems (especially strong ones) without the burden of technical tools and difficulties. An easy way of considering the Wiener process is also given, through the study of the random walk.Since the first and second editions were published in 1990 and 2005, a number of new results have appeared in the literature. The first two editions contained many unsolved problems and conjectures which have since been settled; this third, revised and enlarged edition includes those new results. In this edition, a completely new part is included concerning Simple Random Walks on Graphs. Properties of random walks on several concrete graphs have been studied in the last decade. Some of the obtained results are also presented.

The simplest mathematical model of the Brownian motion of physics is the simple, symmetric random walk. This book collects and compares current results — mostly strong theorems which describe the properties of a random walk. The modern problems of the limit theorems of probability theory are treated in the simple case of coin tossing. Taking advantage of this simplicity, the reader is familiarized with limit theorems (especially strong ones) without the burden of technical tools and difficulties. An easy way of considering the Wiener process is also given, through the study of the random walk.Since the first edition was published in 1990, a number of new results have appeared in the literature. The original edition contained many unsolved problems and conjectures which have since been settled; this second revised and enlarged edition includes those new results. Three new chapters have been added: frequently and rarely visited points, heavy points and long excursions. This new edition presents the most complete study of, and the most elementary way to study, the path properties of the Brownian motion.

The aim of this book is to report on the progress realized in probability theory in the field of dynamic random walks and to present applications in computer science, mathematical physics and finance. Each chapter contains didactical material as well as more advanced technical sections. Few appendices will help refreshing memories (if necessary!). · New probabilistic model, new results in probability theory · Original applications in computer science · Applications in mathematical physics · Applications in finance

The author's previous book, Random Walk in Random and Non-Random Environments, was devoted to the investigation of the Brownian motion of a simple particle. The present book studies the independent motions of infinitely many particles in the d-dimensional Euclidean space Rd. In Part I the particles at time t = 0 are distributed in Rd according to the law of a given random field and they execute independent random walks. Part II is devoted to branching random walks, i.e. to the case where the particles execute random motions and birth and death processes independently. Finally, in Part III, functional laws of iterated logarithms are proved for the cases of independent motions and branching processes.

Classical probability theory provides information about random walks after a fixed number of steps. For applications, however, it is more natural to consider random walks evaluated after a random number of steps. Examples are sequential analysis, queuing theory, storage and inventory theory, insurance risk theory, reliability theory, and the theory of contours. Stopped Random Walks: Limit Theorems and Applications shows how this theory can be used to prove limit theorems for renewal counting processes, first passage time processes, and certain two-dimenstional random walks, and to how these results are useful in various applications. This second edition offers updated content and an outlook on further results, extensions and generalizations. A new chapter examines nonlinear renewal processes in order to present the analagous theory for perturbed random walks, modeled as a random walk plus "noise."

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.

This is the second volume of a two-volume work devoted to probability theory in physical chemistry, and engineering. Rather than dealing explicitly with the idea of an ongoing random walk, with each chaotic step taking place at fixed time intervals, this volume addresses random environments-- models in which the disorder is frozen in space. It begins with an introduction to the geometry of random environments, emphasizing Bernoulli percolation models. The scope of the investigation then widens as we ask how structural disorder affects the transport process. The final chapters confront the interplay of two different forms of randomness; spatial randomness frozen into the environment and temporal randomness associated with the choices for next steps made by a random walker. The book ends with a discussion of "the ant in the labyrinth" problems and an extensive bibliography that, along with the rest of the material, will be of value to researchers in physics, mathematics, and chemical engineering.