The Parabolic Anderson Model
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Author | : Wolfgang König |
Publisher | : Birkhäuser |
Total Pages | : 199 |
Release | : 2016-06-30 |
Genre | : Mathematics |
ISBN | : 3319335960 |
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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Total Pages | : 201 |
Release | : 2014 |
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Author | : Graham Mueller |
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Total Pages | : 202 |
Release | : 2010 |
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Author | : René Carmona |
Publisher | : American Mathematical Soc. |
Total Pages | : 138 |
Release | : 1994 |
Genre | : Mathematics |
ISBN | : 0821825771 |
This book is devoted to the analysis of the large time asymptotics of the solutions of the heat equation in a random time-dependent potential. The authors give complete results in the discrete case of the d-dimensional lattice when the potential is, at each site, a Brownian motion in time. The phenomenon of intermittency of the solutions is discussed.
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Total Pages | : 81 |
Release | : 2011 |
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Author | : Wolfgang König |
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Total Pages | : 15 |
Release | : 2010 |
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Author | : Adrian Schnitzler |
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Total Pages | : 15 |
Release | : 2010 |
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Author | : J. Gärtner |
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Total Pages | : 19 |
Release | : 2005 |
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Author | : Marek Biskup |
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Total Pages | : 37 |
Release | : 2000 |
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Author | : Michael Brian Rael |
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Total Pages | : 49 |
Release | : 2013 |
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ISBN | : 9781303097201 |
In this dissertation we present various results pertaining to the Parabolic Anderson Model. First we show that the Lyapunov exponent, lambda(kappa), of the Parabolic Anderson Model in continuous space with Stratonovich differential is O(kappa1/3 near 0. We prove the required upper bound, the lower bound having been proven in (Cranston & Mountford 2006). Second, we prove the existence of stationary measures for the Parabolic Anderson Model in continuous space with Ito differential. Furthermore, we prove that these measures are associated and determined by the average mass of the initial configuration. Finally we present progress towards computing the Lyapunov exponent of the Quasi-Stationary Parabolic Anderson Model. We prove a smaller upper bound on lambda(kappa), improving on the work in (Boldrighini, Molchanov, & Pellegrinotti 2007), but our bound is not sharp. Computing lambda(kappa) in this model remains an open problem.