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3080 results in Probability theory and stochastic processes

Page 41 of 154

Quantum Fields and Processes
  • A Combinatorial Approach
  • John Gough, Joachim Kupsch
  • Published online:
    23 March 2018
    Print publication:
    12 April 2018
  • Wick ordering of creation and annihilation operators is of fundamental importance for computing averages and correlations in quantum field theory and, by extension, in the Hudson–Parthasarathy theory of quantum stochastic processes, quantum mechanics, stochastic processes, and probability. This book develops the unified combinatorial framework behind these examples, starting with the simplest mathematically, and working up to the Fock space setting for quantum fields. Emphasizing ideas from combinatorics such as the role of lattice of partitions for multiple stochastic integrals by Wallstrom–Rota and combinatorial species by Joyal, it presents insights coming from quantum probability. It also introduces a 'field calculus' which acts as a succinct alternative to standard Feynman diagrams and formulates quantum field theory (cumulant moments, Dyson–Schwinger equation, tree expansions, 1-particle irreducibility) in this language. Featuring many worked examples, the book is aimed at mathematical physicists, quantum field theorists, and probabilists, including graduate and advanced undergraduate students.

Stochastic Dynamics, Filtering and Optimization
  • Debasish Roy, G. Visweswara Rao
  • Published online:
    08 February 2018
    Print publication:
    01 January 2017
  • Targeted at graduate students, researchers and practitioners in the field of science and engineering, this book gives a self-contained introduction to a measure-theoretic framework in laying out the definitions and basic concepts of random variables and stochastic diffusion processes. It then continues to weave into a framework of several practical tools and applications involving stochastic dynamical systems. These include tools for the numerical integration of such dynamical systems, nonlinear stochastic filtering and generalized Bayesian update theories for solving inverse problems and a new stochastic search technique for treating a broad class of non-convex optimization problems. MATLAB® codes for all the applications are uploaded on the companion website.

  • Preface

  • Geoffrey Grimmett, Statistical Laboratory, University of Cambridge
  • Book:
    Probability on Graphs
    Published online:
    12 January 2018
    Print publication:
    25 January 2018, pp ix-xii

  • Summary

    Within themenagerie of objects studied in contemporary probability theory, a number of related animals have attracted great interest amongst probabilists and physicists in recent years. The inspiration for many of these objects comes from physics, but the mathematical subject has taken on a life of its own and many beautiful constructions have emerged. The overall target of these notes is to identify some of these topics, and to develop their basic theory at a level suitable for mathematics graduates.

    If the two principal characters in these notes are random walk and percolation, they are only part of the rich theory of uniform spanning trees, self-avoiding walks, random networks, models for ferromagnetism and the spread of disease, and motion in random environments. This is an area that has attracted many fine scientists, by virtue, perhaps, of its special mixture of modelling and problem-solving. There remain many open problems. It is the experience of the author that these may be explained successfully to a graduate audience open to inspiration and provocation.

    The material described here may be used for personal study and also as the bases of lecture courses of between 16 and 48 hours duration. Little is assumed about the mathematical background of the audience beyond some basic probability theory, but students should be willing to get their hands dirty if they are to profit. Care should be taken in the setting of examinations, since problems can be unexpectedly difficult. Successful examinations may be designed, and some help is offered through the inclusion of exercises at the ends of chapters. As an alternative to a conventional examination, students could be asked to deliver presentations on aspects and extensions of the topics studied.

    Chapter 1 is devoted to the relationship between randomwalks (on graphs) and electrical networks. This leads to the Thomson andRayleigh principles, and thence to a proof of Pólya's theorem. In Chapter 2,we describeWilson's algorithm for constructing a uniform spanning tree (UST), and we discuss boundary conditions and weak limits for UST on a lattice. This chapter includes a brief introduction to Schramm–Löwner evolutions (SLEs).

Page 41 of 154