Potential Functions of Random Walks in with Infinite Variance

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.

Emphasises the significance of the potential function Gives classical proofs of new and established results Generalises old results to new settings

Autorentext
Kôhei Uchiyama is a professor emeritus of mathematics at the Tokyo Institute of Technology, where he obtained his doctorate in science in 1978. After working at Ryukyu University, Nara Women's University, and Hiroshima University, he moved to the Tokyo Institute of Technology in 1998. While working on various problems concerning statistical mechanics, he has continued interest in Brownian motion and random walks throughout his academic career.

Inhalt
Preface.- Introduction.- Preliminaries.- Bounds of the Potential Function.- Some Explicit Asymptotic Forms of a(x).- Applications Under m+/m 0.- The Two-Sided Exit Problem General Case.- The Two-Sided Exit Problem for Relatively Stable Walks.- Absorption Problems for Asymptotically Stable Random Walks.- Asymptotically Stable RandomWalks Killed Upon Hitting a Finite Set.- Appendix.- References.- Notation Index.- Subject Index.<p