Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

Abstract

The parabolic Anderson model is defined as the partial differential equation ∂u(x, t)/∂t = κ u(x, t) + ξ (x, t)u(x, t), x ∈ Z d , t ≥ 0, where κ ∈ [0, ∞) is the diffusion constant, is the discrete Laplacian, and ξ is a dynamic random environment that drives the equation. The initial condition u(x, 0) = u 0 (x), x ∈ Z d , is typically taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate 2dκ, split into two at rate ξ ∨ 0, and die at rate (−ξ) ∨ 0. In earlier work we looked at the Lyapunov exponentsFor the former we derived quantitative results on the κ-dependence for four choices of ξ : space-time white noise, independent simple random walks, the exclusion process and the voter model. For the latter we obtained qualitative results under certain space-time mixing conditions on ξ . In the present paper we investigate what happens when κ is replaced by K , where where, for a fixed realisation of K, Supp(K) is the set of values taken by the Kfield. We also show that for the associated quenched Lyapunov exponent λ 0 (K) this formula only provides a lower bound, and we conjecture that an upper bound holds when Supp(K) is replaced by its convex hull. Our proof is valid for three classes of reversible ξ , and for all K satisfying a certain clustering property, namely, there are arbitrarily large balls where K is almost constant and close to any value in Supp(K). What our result says is that the annealed Lyapunov exponents are controlled by those pockets of K where the conductances are close to the value that maximises the growth in the homogeneous setting. In contrast our conjecture says that the quenched Lyapunov exponent is controlled by a mixture of pockets of K where the conductances are nearly constant. Our proof is based on variational representations and confinement arguments.

Reaction–Diffusion models: From particle systems to SDE’s

“…By the Martingale Representation Theorem (see discussion on section 3) we can write on a convenient probability space that M * t = t 0 α(ζ * s ) l dB s where B · is a Brownian motion. Now, we can finally take limits on (7) and obtain:…”

“…In the parabolic Anderson model, ζ t (x) corresponds to the expected number of particles of a system of independent random walks which split with rate (dB x t ) + and annihilated with rate (dB x t ) − . Although these rates are not well defined, they make sense in a weak sense using Itô's integral, see [7] for a recent reference. In that article, equation (3) is interpreted as a catalytic equation: the noises B x t act as an independent catalyst which favours of penalizes the reproduction of the individuals.…”

The Parabolic Anderson Model

“…The PAM with RWRC as the underlying random motion has not yet been studied (however, note the work [ErhHolMai15b] on the case of a time-dependent potential with a RWRC with uniformly elliptic random conductances, see Remark 8.7), but important prerequisities have been derived: annealed large-deviation principles (LDPs) for the normalised local times of the RWRC in fixed boxes [KönSalWol12] and in time-depending, growing boxes [KönWol15]; see also the thesis [Wol13]. These LDPs are particularly interesting because of the assumption on the conductances that they are not uniformly elliptic (i.e., not bounded away from zero and from infinity), but can attain arbitrarily small values.…”

“…In [ErhHolMai15b], the simple random walk is replaced by the random walk among random conductances (RWRC), i.e., the operator Ä d is replaced by the randomised Laplace operator d ! f .z/ D P x z !…”