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.
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