The Parabolic Anderson Model

Random Walk in Random Potential

Wolfgang König (Autor)

Buch | Softcover

XI, 192 Seiten

2018 | 1. Softcover reprint of the original 1st ed. 2016
Springer International Publishing (Verlag)
978-3-319-81556-5 (ISBN)

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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.

1 Background, model and questions.- 2 Tools and concepts.- 3 Moment asymptotics for the total mass.- 4 Some proof techniques.- 5 Almost sure asymptotics for the total mass.- 6 Strong intermittency.- 7 Refined questions.- 8 Time-dependent potentials.

"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)

“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)

Erscheinungsdatum	05.03.2022
Reihe/Serie	Pathways in Mathematics
Zusatzinfo	XI, 192 p. 4 illus.
Verlagsort	Cham
Sprache	englisch
Maße	155 x 235 mm
Gewicht	3168 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Wahrscheinlichkeit / Kombinatorik
Themenwelt	Naturwissenschaften ► Physik / Astronomie
Schlagworte	Anderson localization • Eigenvalue order statistics • Feynman-Kac formula • Heat equation with random coefficients • intermittency • large deviations • Mass concentration • Random Schrödinger operator • Random walk in random potential
ISBN-10	3-319-81556-3 / 3319815563
ISBN-13	978-3-319-81556-5 / 9783319815565
Zustand	Neuware