© 2016

The Parabolic Anderson Model

Random Walk in Random Potential


  • This book omits all technicalities and comes quickly to the point

  • This volume organizes an enormous wealth of material in a quickly understandable way

  • It reports on the latest achievements in an active and timely research field on the edge between probability and analysis that has a lot of cross-connections


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


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.


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

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

About the authors

Bibliographic information

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“A visible feature of the book is its emphasis on the tool and idea development. … The book is particularly beneficial to the following two groups’ of readers: (1) Those who have been working in the direction of PAM or related areas … (2) Young researchers who are considering starting their work in this direction. … Remarks and examples are distributed throughout the entire book with historic accounts and specific discussions, which further increases the readability of the book.” (Xia Chen, Mathematical Reviews, June, 2017)