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

摘要

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 equationdescribes 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 thesuccessive moments of $u$ w.r.t.\ $\xi$, was investigated. It was shown thatthese exponents exhibit an interesting dependence on the dimension and on thediffusion constant. In the present paper we address some questions left open inG\"artner, den Hollander and Maillard 2010 by considering specifically when theLyapunov exponents are the a priori maximal value in terms of strong transienceof the Markov process underlying the voter model.