Parabolic Anderson model with voter catalysts: dichotomy in the behavior of Lyapunov exponents

TL;DR

This paper investigates the behavior of Lyapunov exponents in the parabolic Anderson model influenced by voter catalysts, revealing a dichotomy based on the transience properties of the underlying voter process.

Contribution

It specifically analyzes when Lyapunov exponents reach their maximal value, linking this to the strong transience of the voter model's Markov process.

Findings

Lyapunov exponents reach maximum under strong transience conditions.

Behavior depends on the dimension and diffusion constant.

Provides conditions for maximal growth rates of the solution.

Abstract

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.