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The Parabolic Anderson Model
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Book Synopsis The Parabolic Anderson Model by : Wolfgang König
Download or read book The Parabolic Anderson Model written by Wolfgang König and published by Birkhäuser. This book was released on 2016-06-30 with total page 199 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Parabolic Anderson Model and Long-range Percolation by :
Download or read book The Parabolic Anderson Model and Long-range Percolation written by and published by . This book was released on 2014 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Association and Dependence with Applications to the Parabolic Anderson Model by : Graham Mueller
Download or read book Association and Dependence with Applications to the Parabolic Anderson Model written by Graham Mueller and published by . This book was released on 2010 with total page 202 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Parabolic Anderson Problem and Intermittency by : René Carmona
Download or read book Parabolic Anderson Problem and Intermittency written by René Carmona and published by American Mathematical Soc.. This book was released on 1994 with total page 138 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis The Parabolic Anderson Model with Acceleration and Deceleration by : Wolfgang König
Download or read book The Parabolic Anderson Model with Acceleration and Deceleration written by Wolfgang König and published by . This book was released on 2010 with total page 15 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Asymptotic Properties of the Parabolic Anderson Model by :
Download or read book Asymptotic Properties of the Parabolic Anderson Model written by and published by . This book was released on 2011 with total page 81 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Precise Asymptotics for the Parabolic Anderson Model with a Moving Catalyst Or Trap by : Adrian Schnitzler
Download or read book Precise Asymptotics for the Parabolic Anderson Model with a Moving Catalyst Or Trap written by Adrian Schnitzler and published by . This book was released on 2010 with total page 15 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Annealed Asymptotics for the Parabolic Anderson Model with a Moving Catalyst by : J. Gärtner
Download or read book Annealed Asymptotics for the Parabolic Anderson Model with a Moving Catalyst written by J. Gärtner and published by . This book was released on 2005 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Long Time Tails in the Parabolic Anderson Model with Bounded Potential by : Marek Biskup
Download or read book Long Time Tails in the Parabolic Anderson Model with Bounded Potential written by Marek Biskup and published by . This book was released on 2000 with total page 37 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Results on the Parabolic Anderson Model by : Michael Brian Rael
Download or read book Results on the Parabolic Anderson Model written by Michael Brian Rael and published by . This book was released on 2013 with total page 49 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Book Synopsis Correlation Structure of Intermittency in the Parabolic Anderson Model by : J. Gärtner
Download or read book Correlation Structure of Intermittency in the Parabolic Anderson Model written by J. Gärtner and published by . This book was released on 1997 with total page 60 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis Parabolic Anderson Model with Finite Number of Moving Catalysts by : Fabienne Castell
Download or read book Parabolic Anderson Model with Finite Number of Moving Catalysts written by Fabienne Castell and published by . This book was released on 2011 with total page 24 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Parabolic Anderson Model on Galton-Watson Trees by :
Download or read book The Parabolic Anderson Model on Galton-Watson Trees written by and published by . This book was released on 2024 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Book Synopsis The Parabolic Anderson Model on the Hypercube by : Luca Avena
Download or read book The Parabolic Anderson Model on the Hypercube written by Luca Avena and published by . This book was released on 2016 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We consider the parabolic Anderson model (PAM) on the n-dimensional hypercube with random i.i.d. potentials. We parametrize time by volume and study the solution at the location of the k-th largest potential. Our main result is that, for a certain class of potential distributions, the solution exhibits a phase transition: for short time scales it behaves like a system without diffusion, whereas, for long time scales the growth is dictated by the principle eigenvalue and the corresponding eigenfunction of the Anderson operator, for which we give precise asymptotics. Moreover, the transition time depends only on the difference between the largest and the k-th largest potential. One of our main motivations in this article is to investigate the mutationselection model of population genetics on a random fitness landscape, which is given by the ratio of the solution of PAM to its total mass, with the field corresponding to the fitness landscape. We show that the phase transition of the solution translates to the mutation-selection model as follows: a population initially concentrated at the site of the k-th best fitness value moves completely to the site of the best fitness on time scales where the transition of growth rates happens. The class of potentials we consider involve the Random Energy Model (REM) of statistical physics which is studied as one of the main examples of a random fitness landscape.
Book Synopsis On Global Solutions of the Parabolic Anderson Model and Directed Polymers in Random Environment by : Beatriz Carolina Navarro Lameda
Download or read book On Global Solutions of the Parabolic Anderson Model and Directed Polymers in Random Environment written by Beatriz Carolina Navarro Lameda and published by . This book was released on 2020 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: This thesis studies global solutions to the semidiscrete stochastic heat equation and the associated Cauchy problem known as Parabolic Anderson Model. Via a Feynman-Kac formula, it is linked with the analysis of directed polymers in random environment, and this thesis establishes a number of results for the corresponding partition function. We consider a continuous-time simple symmetric random walk on the integer lattice $\Z^d$ in dimension $d \geq 3$, subject to a random potential given by two-sided Wiener processes. In the high-temperature regime, we prove the existence of the $L^2$- and almost sure limit of the partition function as time $t \to \pm \infty$. We show that the $L^2$-convergence rate is at least polynomial and that the limiting partition function is positive almost surely. Furthermore, we show that this limiting partition function defines a global stationary solution to the semidiscrete stochastic heat equation which is unique up to a rescaling, and which in some sense attracts solutions to the Parabolic Anderson Model for any subexponentially growing initial data. One of the primary tools in the proof of this uniqueness and attraction result is a factorization formula for the point-to-point partition function, which is related to the ones obtained by Sinai (1995) and Kifer (1997) for other polymer models, but valid not only on the diffusive scale but up to any sub-ballistic scale. This factorization formula allows us to obtain a uniqueness result for physical invariant probability measures of a certain skew product that can be naturally associated with the semidiscrete stochastic heat equation, which in turns gives uniqueness of global stationary solutions.
Book Synopsis The Parabolic Anderson Model on a Galton-Watson Tree by : Frank den Hollander
Download or read book The Parabolic Anderson Model on a Galton-Watson Tree written by Frank den Hollander and published by . This book was released on 2020 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: We study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence.
Book Synopsis Moment Asymptotics for the Continuous Parabolic Anderson Model by : Jürgen Gärtner
Download or read book Moment Asymptotics for the Continuous Parabolic Anderson Model written by Jürgen Gärtner and published by . This book was released on 1998 with total page 27 pages. Available in PDF, EPUB and Kindle. Book excerpt:
