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The Parabolic Anderson Model

Random Walk in Random Potential

  • Wolfgang König

Part of the Pathways in Mathematics book series (PATHMATH)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Wolfgang König
    Pages 1-18
  3. Wolfgang König
    Pages 19-41
  4. Wolfgang König
    Pages 43-70
  5. Wolfgang König
    Pages 71-84
  6. Wolfgang König
    Pages 85-97
  7. Wolfgang König
    Pages 99-122
  8. Wolfgang König
    Pages 123-157
  9. Wolfgang König
    Pages 159-171
  10. Back Matter
    Pages 173-192

About this book

Introduction

This is a comprehensive survey on the research on the parabolic Anderson model – the heat equation with random potential or the random walk in random potential – of the years 1990 – 2015. The investigation of this model requires a combination of tools from probability (large deviations, extreme-value theory, e.g.) and analysis (spectral theory for the Laplace operator with potential, variational analysis, e.g.). We explain the background, the applications, the questions and the connections with other models and formulate the most relevant results on the long-time behavior of the solution, like quenched and annealed asymptotics for the total mass, intermittency, confinement and concentration properties and mass flow. Furthermore, we explain the most successful proof methods and give a list of open research problems. Proofs are not detailed, but concisely outlined and commented; the formulations of some theorems are slightly simplified for better comprehension.

Keywords

Random walk in random potential Heat equation with random coefficients Random Schrödinger operator Feynman-Kac formula Anderson localization Mass concentration Intermittency Large deviations Eigenvalue order statistics

Authors and affiliations

  • Wolfgang König
    • 1
  1. 1.für Angewandte Ana. und Stoc.;TU BerlinWeierstraß-Institut;Institute for MathemBerlinGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-33596-4
  • Copyright Information Springer International Publishing Switzerland 2016
  • Publisher Name Birkhäuser, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-33595-7
  • Online ISBN 978-3-319-33596-4
  • Series Print ISSN 2367-3451
  • Series Online ISSN 2367-346X
  • Buy this book on publisher's site