Multiparameter processes extend the existing one-parameter theory of random processes in an elegant way, and have found connections to diverse disciplines such as probability theory, real and functional analysis, group theory, analytic number theory, and group renormalization in mathematical physics, to name a few.This book lays the foundation of aspects of the rapidly-developing subject of random fields, and is designed for a second graduate course in probability and beyond. Its intended audience is pure, as well as applied, mathematicians.Davar Khoshnevisan is Professor of Mathematics at the University of Utah. His research involves random fields, probabilistic potential theory, and stochastic analysis. Self-contained presentation: from elementary material to state-of-the-art research; Much of the theory in book-form for the first time; Connections are made between probability and other areas of mathematics, engineering and mathematical physics
DISCRETE-PARAMETER RANDOM FIELDS Discrete Parameter Martingales Two Applications in Analysis Random Walks Multiparameter Walks Gaussian Random Walks Limit Theorems; CONTINUOUS-PARAMETER RANDOM FIELDS Continuous Parameter Martingales Constructing Markov Processes Generation of Markov Processes Probabilistic Potential Theory Multiparameter Markov Processes Brownian Sheet and Potential Theory; APPENDICES Kolmogorov's Consistency Theorem Laplace Transforms Hausdorff Dimensions and Measures Energy and Capacity
From the reviews:
"This book presents an updated and comprehensible account on the theory of multiparameter stochastic processes. ... This book is certainly a basic reference for subjects like multiparameter martingales and potential theory for the Brownian sheet and several Markov processes. It can be useful for researchers who would like to learn the basis and recent developments of these subjects. The book is self-contained ... . In spite of the technical character of the subject, reading this book is a very pleasant and enriching experience." (David Nualart, Mathematical Reviews, Issue 2004 a)
"This book aims to construct a general framework for the analysis of a large class of random fields, also known as multiparameter processes. A great part of one-parameter theory is also included, with the goal to keep the book self-contained. ... The book contains a lot of supplementary exercises, extended theoretical appendices and is useful both for highly qualified specialists and for advanced graduate students." (Yu. S. Mishura, Zentralblatt Math, Vol. 1005, 2003)
"The present book is the first one presenting a general treatment of random fields. ... The book can be recommended not only to probabilists but to anyone interested in the applications of probability within other areas of mathematics, particularly in analysis." (P. Révész, Internationale Mathematische Nachrichten, Vol. 57 (192), 2003)
Self-contained presentation: from elementary material to state-of-the-art research; Much of the theory in book-form for the first time; Connections are made between probability and other areas of mathematics, engineering and mathematical physics
Multi-parameter processes extend the existing one-parameter theory in an elegant way and have many applications to other fields in mathematics. This book on random fields is designed for a second graduate course in probability. Recent work on random fields has made it possible to make it an expository subject which interacts with several other areas in mathematics and has enough mathematical depth to be of use to pure as well as applied mathematicians of many backgrounds.