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Author: Kôhei Uchiyama Publisher: Springer Nature ISBN: 3031410203 Category : Electronic books Languages : en Pages : 277
Book Description
This book studies the potential functions of one-dimensional recurrent random walks on the lattice of integers with step distribution of infinite variance. The central focus is on obtaining reasonably nice estimates of the potential function. These estimates are then applied to various situations, yielding precise asymptotic results on, among other things, hitting probabilities of finite sets, overshoot distributions, Green functions on long finite intervals and the half-line, and absorption probabilities of two-sided exit problems. The potential function of a random walk is a central object in fluctuation theory. If the variance of the step distribution is finite, the potential function has a simple asymptotic form, which enables the theory of recurrent random walks to be described in a unified way with rather explicit formulae. On the other hand, if the variance is infinite, the potential function behaves in a wide range of ways depending on the step distribution, which the asymptotic behaviour of many functionals of the random walk closely reflects. In the case when the step distribution is attracted to a strictly stable law, aspects of the random walk have been intensively studied and remarkable results have been established by many authors. However, these results generally do not involve the potential function, and important questions still need to be answered. In the case where the random walk is relatively stable, or if one tail of the step distribution is negligible in comparison to the other on average, there has been much less work. Some of these unsettled problems have scarcely been addressed in the last half-century. As revealed in this treatise, the potential function often turns out to play a significant role in their resolution. Aimed at advanced graduate students specialising in probability theory, this book will also be of interest to researchers and engineers working with random walks and stochastic systems.
Author: Kôhei Uchiyama Publisher: Springer Nature ISBN: 3031410203 Category : Electronic books Languages : en Pages : 277
Book Description
This book studies the potential functions of one-dimensional recurrent random walks on the lattice of integers with step distribution of infinite variance. The central focus is on obtaining reasonably nice estimates of the potential function. These estimates are then applied to various situations, yielding precise asymptotic results on, among other things, hitting probabilities of finite sets, overshoot distributions, Green functions on long finite intervals and the half-line, and absorption probabilities of two-sided exit problems. The potential function of a random walk is a central object in fluctuation theory. If the variance of the step distribution is finite, the potential function has a simple asymptotic form, which enables the theory of recurrent random walks to be described in a unified way with rather explicit formulae. On the other hand, if the variance is infinite, the potential function behaves in a wide range of ways depending on the step distribution, which the asymptotic behaviour of many functionals of the random walk closely reflects. In the case when the step distribution is attracted to a strictly stable law, aspects of the random walk have been intensively studied and remarkable results have been established by many authors. However, these results generally do not involve the potential function, and important questions still need to be answered. In the case where the random walk is relatively stable, or if one tail of the step distribution is negligible in comparison to the other on average, there has been much less work. Some of these unsettled problems have scarcely been addressed in the last half-century. As revealed in this treatise, the potential function often turns out to play a significant role in their resolution. Aimed at advanced graduate students specialising in probability theory, this book will also be of interest to researchers and engineers working with random walks and stochastic systems.
Author: Wolfgang Woess Publisher: Cambridge University Press ISBN: 0521552923 Category : Mathematics Languages : en Pages : 350
Book Description
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: Gregory F. Lawler Publisher: American Mathematical Soc. ISBN: 0821848291 Category : Mathematics Languages : en Pages : 170
Book Description
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.
Author: Gregory F. Lawler Publisher: Cambridge University Press ISBN: 1139488767 Category : Mathematics Languages : en Pages : 377
Book Description
Random walks are stochastic processes formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice. It is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.
Author: Mikhail Menshikov Publisher: Cambridge University Press ISBN: 1316867366 Category : Mathematics Languages : en Pages : 385
Book Description
Stochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the 'semimartingale' or 'Lyapunov function' method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototypical near-critical systems.
Author: D N Shanbhag Publisher: Gulf Professional Publishing ISBN: 9780444500144 Category : Mathematics Languages : en Pages : 990
Book Description
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.
Author: George H. Weiss Publisher: SIAM ISBN: 0898713234 Category : Science Languages : en Pages : 266
Book Description
This collection of independent articles describes some mathematical problems recently developed in statistical physics and theoretical chemistry. The book introduces and reviews current research on such topics as nonlinear systems and colored noise, stochastic resonance, percolation, the trapping problem in the theory of random walks, and diffusive models for chemical kinetics. Some of these topics have never before been presented in expository book form. Applied mathematicians will be introduced to some contemporary problems in statistical physics. In addition, a number of unsolved problems currently attracting intensive research efforts are described.
Author: Yves Benoist Publisher: Springer ISBN: 3319477218 Category : Mathematics Languages : en Pages : 319
Book Description
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.
Author: A. Tarantola Publisher: Elsevier ISBN: 0444599673 Category : Science Languages : en Pages : 634
Book Description
Inverse Problem Theory is written for physicists, geophysicists and all scientists facing the problem of quantitative interpretation of experimental data. Although it contains a lot of mathematics, it is not intended as a mathematical book, but rather tries to explain how a method of acquisition of information can be applied to the actual world.The book provides a comprehensive, up-to-date description of the methods to be used for fitting experimental data, or to estimate model parameters, and to unify these methods into the Inverse Problem Theory. The first part of the book deals with discrete problems and describes Maximum likelihood, Monte Carlo, Least squares, and Least absolute values methods. The second part deals with inverse problems involving functions.The book is almost completely self-contained, with all important concepts carefully introduced. Although theoretical concepts are strongly emphasized, the author has ensured that all the useful formulas are listed, with many special cases included. The book will thus serve equally well as a reference manual for researchers needing to refresh their memories on a given algorithm, or as a textbook in a course for undergraduate or graduate students.
Author: Kôhei Uchiyama
Author: Kôhei Uchiyama
Author: Wolfgang Woess
Author: Gregory F. Lawler
Author: Gregory F. Lawler
Author: Mikhail Menshikov
Author: D N Shanbhag
Author: George H. Weiss
Author: 
Author: Yves Benoist
Author: A. Tarantola