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Probability Theory and Applications
Edited by: Elton P. Hsu, Northwestern University, Evanston, IL, and S. R. S. Varadhan, New York University-Courant Institute of Mathematical Sciences, NY
A co-publication of the AMS and IAS/Park City Mathematics Institute.
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IAS/Park City Mathematics Series
1999; 374 pp; hardcover
Volume: 6
ISBN-10: 0-8218-0590-8
ISBN-13: 978-0-8218-0590-9
List Price: US$65
Member Price: US$52
Order Code: PCMS/6
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See also:

Statistical Mechanics - Scott Sheffield and Thomas Spencer

This volume, with contributions by leading experts in the field, is a collection of lecture notes of the six minicourses given at the IAS/Park City Summer Mathematics Institute. It introduces advanced graduates and researchers in probability theory to several of the currently active research areas in the field. Each course is self-contained with references and contains basic materials and recent results. Topics include interacting particle systems, percolation theory, analysis on path and loop spaces, and mathematical finance.

The volume gives a balanced overview of the current status of probability theory. An extensive bibliography for further study and research is included. This unique collection presents several important areas of current research and a valuable survey reflecting the diversity of the field.

Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Readership

Advanced graduate students and researchers in probability theory; physicists and researchers in finance and economics.

Table of Contents

  • Introduction
Stochastic spatial models
  • Introduction
  • The voter model
  • Coalescing random walks
  • Voter model with mutation
  • The block construction
  • Long range limits
  • Rapid stirring limits
  • Bibliography
Independent and dependent percolation
  • Preface
  • The basics of percolation
  • Rescaling and finite-size scaling in percolation
  • Critical exponent inequalities
  • Two fundamental questions
  • Finite-size scaling and the incipient infinite cluster
  • The BK(R) inequality
  • The Potts model and the random cluster model
  • Bibliography
Hydrodynamical scaling limits of simple exclusion models
  • Introduction
  • The simple exclusion model
  • Proof of Theorem 1.4
  • Local ergodicity
  • Two-block estimate
  • Relative entropy
  • The Green-Kubo formula and asymmetric simple exclusion processes
  • Some open problems
  • Bibliography
An introduction to analysis on path space
  • Introduction
  • Gaussian measures on a Hilbert space
  • Rolling on
  • About \(\mathcal{W}_M\)
  • A few facts, and something else
  • Bibliography
Analysis on path and loop spaces
  • Introduction
  • Euclidean Brownian motion
  • Gradient operator
  • Ornstein-Uhlenbeck operator
  • Brownian motion on manifolds
  • Gradient formulas
  • Integration by parts
  • Logarithmic Sobolev inequalities
  • Bibliographical comments
  • Bibliography
An introduction to option pricing and the mathematical theory of risk
  • Introduction
  • An introduction to option pricing and the mathematical theory of risk
  • Bibliography
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