Mathematics – Probability
Scientific paper
2010-10-23
Mathematics
Probability
In honour of J\"urgen G\"artner on the occasion of his 60th birthday, 33 pages. Updated version following the referee's commen
Scientific paper
We consider the parabolic Anderson model $\partial u/\partial t = \kappa\Delta u + \gamma\xi u$ with $u\colon\, \Z^d\times R^+\to \R^+$, where $\kappa\in\R^+$ is the diffusion constant, $\Delta$ is the discrete Laplacian, $\gamma\in\R^+$ is the coupling constant, and $\xi\colon\,\Z^d\times \R^+\to\{0,1\}$ is the voter model starting from Bernoulli product measure $\nu_{\rho}$ with density $\rho\in (0,1)$. The solution of this equation describes the evolution of a "reactant" $u$ under the influence of a "catalyst" $\xi$. In G\"artner, den Hollander and Maillard 2010 the behavior of the \emph{annealed} Lyapunov exponents, i.e., the exponential growth rates of the successive moments of $u$ w.r.t.\ $\xi$, was investigated. It was shown that these exponents exhibit an interesting dependence on the dimension and on the diffusion constant. In the present paper we address some questions left open in G\"artner, den Hollander and Maillard 2010 by considering specifically when the Lyapunov exponents are the a priori maximal value in terms of strong transience of the Markov process underlying the voter model.
Maillard Grégory
Mountford Thomas
Schöpfer Samuel
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