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 Electronic Journal of Probability > Vol. 12 (2007) > Paper 18 open journal systems 


Intermittency on catalysts: symmetric exclusion

Jürgen Gärtner, Institut für Mathematik, Technische Universität Berlin
Frank den Hollander, Mathematical Institute, Leiden University
Gregory Maillard, Institut de Mathematiques, Ecole Polytechnique Federale de Lausanne


Abstract
We continue our study of intermittency for the parabolic Anderson equation, i.e., the spatially discrete heat equation on the d-dimensional integer lattice with a space-time random potential. The solution of the equation describes the evolution of a "reactant" under the influence of a "catalyst".
In this paper we focus on the case where the random field is an exclusion process with a symmetric random walk transition kernel, starting from Bernoulli equilibrium. We consider the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the solution. We show that these exponents are trivial when the random walk is recurrent, but display an interesting dependence on the diffusion constant when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents when the diffusion constant tends to infinity, which is controlled by moderate deviations of the random field requiring a delicate expansion argument.
In Gärtner and den Hollander [10] the case of a Poisson field of independent (simple) random walks was studied. The two cases show interesting differences and similarities. Throughout the paper, a comparison of the two cases plays a crucial role.


Full text: PDF

Pages: 516-573

Published on: May 1, 2007
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Electronic Journal of Probability. ISSN: 1083-6489