Parabolic Anderson Model with Voter Catalysts: Dichotomy in the Behavior of Lyapunov Exponents

Abstract

We consider the parabolic Anderson model ∂u∕∂t=κΔu+γξu with u:${\mathbb{Z}}^{d} \times{\mathbb{R}}^{+} \rightarrow{\mathbb{R}}^{+}$, where $\kappa\in{\mathbb{R}}^{+}$ is the diffusion constant, Δ is the discrete Laplacian, $\gamma\in{\mathbb{R}}^{+}$ is the coupling constant, and $\xi : \,{\mathbb{Z}}^{d} \times{\mathbb{R}}^{+} \rightarrow \{ 0,1\}$ is the voter model starting from Bernoulli product measure ν_ρ with density ρ∈(0,1). The solution of this equation describes the evolution of a “reactant” u under the influence of a “catalyst” ξ.

In Gärtner, den Hollander and Maillard [Gärtner et al., Intermittency on catalysts: Voter model. Ann. Probab. 38, 2066–2102 (2010)] the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of u w.r.t. ξ, 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 [Gärtner et al., Intermittency on catalysts: Voter model. Ann. Probab. 38, 2066–2102 (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.