Abstract
We continue our study of intermittency for the parabolic Anderson equation, i.e., the spatially discrete heat equation on the d-dimensional integer lattice with a space-time random potential. The solution of the equation describes the evolution of a "reactant" under the influence of a "catalyst". <br> In this paper we focus on the case where the random field is an exclusion process with a symmetric random walk transition kernel, starting from Bernoulli equilibrium. We consider the annealed Lyapunov exponents, i.e., the exponential growth rates of the successive moments of the solution. We show that these exponents are trivial when the random walk is recurrent, but display an interesting dependence on the diffusion constant when the random walk is transient, with qualitatively different behavior in different dimensions. Special attention is given to the asymptotics of the exponents when the diffusion constant tends to infinity, which is controlled by moderate deviations of the random field requiring a delicate expansion argument. <br> In Gärtner and den Hollander [10] the case of a Poisson field of independent (simple) random walks was studied. The two cases show interesting differences and similarities. Throughout the paper, a comparison of the two cases plays a crucial role.
Highlights
Introduction and main resultsThe parabolic Anderson equation is the partial differential equation ∂ ∂t u(x, t) = κ∆u(x, t) +ξ(x, t)u(x, t), x ∈ Zd, t ≥ 0. (1.1.1)Here, the u-field is R-valued, κ ∈ [0, ∞) is the diffusion constant, ∆ is the discrete Laplacian, acting on u as ∆u(x, t) =[u(y, t) − u(x, t)] (1.1.2)
Where {Wx, x ∈ Zd} denotes a collection of independent Brownian motions. (In this important case, equation (1.1.1) corresponds to an infinite system of coupledIto diffusions.) They showed that for d = 1, 2 intermittency of all orders is present for all κ, whereas for d ≥ 3 p-intermittency holds if and only if the diffusion constant κ is smaller than a critical threshold κ∗p tending to infinity as p → ∞
If the catalyst is driven by a recurrent random walk, it suffers from “traffic jams”, i.e., with not too small a probability there is a large region around the origin that the catalyst fully occupies for a long time
Summary
(In this important case, equation (1.1.1) corresponds to an infinite system of coupledIto diffusions.) They showed that for d = 1, 2 intermittency of all orders is present for all κ, whereas for d ≥ 3 p-intermittency holds if and only if the diffusion constant κ is smaller than a critical threshold κ∗p tending to infinity as p → ∞ They studied the asymptotics of the quenched Lyapunov exponent in the limit as κ ↓ 0, which turns out to be singular. We choose ξ(·, 0) according to the Bernoulli product measure with density ρ ∈ (0, 1), i.e., initially each site has a particle with probability ρ and no particle with probability 1 − ρ, independently for different sites For this choice, the ξ-field is stationary in time. This powerful inequality will allow us to obtain bounds that are more computable
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
