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3044 results in Probability theory and stochastic processes

Page 55 of 153

Stochastic Analysis
  • Itô and Malliavin Calculus in Tandem
  • Hiroyuki Matsumoto, Setsuo Taniguchi
  • Published online:
    17 November 2016
    Print publication:
    07 November 2016
  • Thanks to the driving forces of the Itô calculus and the Malliavin calculus, stochastic analysis has expanded into numerous fields including partial differential equations, physics, and mathematical finance. This book is a compact, graduate-level text that develops the two calculi in tandem, laying out a balanced toolbox for researchers and students in mathematics and mathematical finance. The book explores foundations and applications of the two calculi, including stochastic integrals and differential equations, and the distribution theory on Wiener space developed by the Japanese school of probability. Uniquely, the book then delves into the possibilities that arise by using the two flavors of calculus together. Taking a distinctive, path-space-oriented approach, this book crystallizes modern day stochastic analysis into a single volume.

Gaussian Processes on Trees
  • From Spin Glasses to Branching Brownian Motion
  • Anton Bovier
  • Published online:
    17 November 2016
    Print publication:
    07 November 2016
  • Branching Brownian motion (BBM) is a classical object in probability theory with deep connections to partial differential equations. This book highlights the connection to classical extreme value theory and to the theory of mean-field spin glasses in statistical mechanics. Starting with a concise review of classical extreme value statistics and a basic introduction to mean-field spin glasses, the author then focuses on branching Brownian motion. Here, the classical results of Bramson on the asymptotics of solutions of the F-KPP equation are reviewed in detail and applied to the recent construction of the extremal process of BBM. The extension of these results to branching Brownian motion with variable speed are then explained. As a self-contained exposition that is accessible to graduate students with some background in probability theory, this book makes a good introduction for anyone interested in accessing this exciting field of mathematics.

  • Preface

    • By Anton Bovier, Rheinische Friedrich-Wilhelms-Universität Bonn
  • Anton Bovier, Rheinische Friedrich-Wilhelms-Universität Bonn
  • Book:
    Gaussian Processes on Trees
    Published online:
    17 November 2016
    Print publication:
    07 November 2016, pp vii-ix

  • Summary

    The title of this book is owed in large part to my personal motivation to study the material I present here. It is rooted in the problem of so-called mean-field models of spin glasses. I will not go into a discussion of the physical background of these systems (see, e.g., [25]). The key mathematical objects associated with them are random functions (called Hamiltonians on some highdimensional space, e.g. ﹛−1, 1﹜n . The standard model here is the Sherrington– Kirkpatrick model, introduced in a seminal paper [103] in 1972. Here the Hamiltonian can be seen as a Gaussian process indexed by the hypercube ﹛−1, 1﹜n whose covariance is a function of the Hamming distance. The attempt to understand the structure of these objects has given rise to the remarkable heuristic theory of replica symmetry breaking developed by Parisi and collaborators (see the book [91]). A rigorous mathematical corroboration of this theory was obtained only rather recently through the work of Talagrand [109, 108, 110, 111], Guerra [63], Aizenman, Simms and Starr [6] and Panchenko [97], to name the most important ones.

    A second class of models that are significantly more approachable by rigorous mathematics was introduced by Derrida and Gardner [46, 59]. Here the Hamming distance was replaced by the lexicographic ultra-metric on ﹛−1, 1﹜. The resulting class of models are called the generalised random energy models (GREM). These processes can be realised as branching random walks with Gaussian increments and thus provide the link to the general topic of this book.

    From branching random walks it is a small step to branching Brownian motion (BBM), a classical object of probability theory, introduced by Moyal [1, 92] in 1962. BBM has been studied over the last 50 years as a subject of interest in its own right, with seminal contributions by McKean [90], Bramson [33, 34], Lalley and Sellke [83], Chauvin and Rouault [39, 40] and others. Recently, the field has experienced a revival with many remarkable contributions and repercussions in other areas.

Page 55 of 153