The Parabolic Anderson Model: Random Walk in Random Potential

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Birkhäuser, Jun 30, 2016 - Mathematics - 192 pages

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.

 

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Contents

1 Background Model and Questions
1
2 Tools and Concepts
19
3 Moment Asymptotics for the Total Mass
42
4 Some Proof Techniques
71
5 Almost Sure Asymptotics for the Total Mass
85
6 Details About Intermittency
99
7 Refined Questions
123
8 TimeDependent Potentials
158
Open Problems
173
Bibliography
178
Index
189
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