Abstract
We continue our study of intermittency for the parabolic Anderson model $\partial u/\partial t = \kappa\Delta u + \xi u$ in a space-time random medium $\xi$, where $\kappa$ is a positive diffusion constant, $\Delta$ is the lattice Laplacian on $\mathbb{Z}^d$, $d \geq 1$, and $\xi$ is a simple symmetric exclusion process on $\mathbb{Z}^d$ in Bernoulli equilibrium. This model describes the evolution of a reactant $u$ under the influence of a catalyst $\xi$. <br /><br /> In Gärtner, den Hollander and Maillard [3] we investigated the behavior of the annealed Lyapunov exponents, i.e., the exponential growth rates as $t\to\infty$ of the successive moments of the solution $u$. This led to an almost complete picture of intermittency as a function of $d$ and $\kappa$. In the present paper we finish our study by focussing on the asymptotics of the Lyaponov exponents as $\kappa\to\infty$ in the critical dimension $d=3$, which was left open in Gärtner, den Hollander and Maillard [3] and which is the most challenging. We show that, interestingly, this asymptotics is characterized not only by a Green term, as in $d\geq 4$, but also by a polaron term. The presence of the latter implies intermittency of all orders above a finite threshold for $\kappa$.
Highlights
Introduction and main resultIn this paper we consider the parabolic Anderson model (PAM) on d, d ≥ 1,∂ u = κ∆u + ξu on d × [0, ∞), u∂(t·, 0) = 1 on d, (1.1)where κ is a positive diffusion constant, ∆ is the lattice Laplacian acting on u as ∆u(x, t) =[u( y, t) − u(x, t)] y∈ d y−x =1 (1.2)( · is the Euclidian norm), and ξ =t≥0, ξt = {ξt (x) : x ∈ d }, (1.3)is a space-time random field that drives the evolution
Where κ is a positive diffusion constant, ∆ is the lattice Laplacian acting on u as
If ξ is given by an infinite particle system dynamics, the solution u of the PAM may be interpreted as the concentration of a diffusing reactant under the influence of a catalyst performing such a dynamics
Summary
Where κ is a positive diffusion constant, ∆ is the lattice Laplacian acting on u as. is a space-time random field that drives the evolution. If ξ is given by an infinite particle system dynamics, the solution u of the PAM may be interpreted as the concentration of a diffusing reactant under the influence of a catalyst performing such a dynamics. In Gärtner, den Hollander and Maillard [3] we studied the PAM for ξ Symmetric Exclusion (SE), and developed an almost complete qualitative picture. We restrict to Simple Symmetric Exclusion (SSE), i.e., (ξt )t≥0 is the Markov dynamics on Ω = {0, 1} 3 (0 = vacancy, 1 = particle) with generator L acting on cylinder functions f : Ω → as. (See Liggett [7], Chapter VIII.) Let η and η denote probability and expectation for ξ given ξ0 = η ∈ Ω. The probability measures νρ, ρ ∈ (0, 1), are the only extremal equilibria of the SSE dynamics.
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