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Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances

Content Provider	Springer Nature Link
Author	Hollander, F. Maillard, G. Erhard, D.
Copyright Year	2016
Abstract	The parabolic Anderson model is defined as the partial differential equation ∂ u(x, t)/∂ t = κ Δ u(x, t) + ξ(x, t)u(x, t), x ∈ ℤ$^{ d }$, t ≥ 0, where κ ∈ [0, ∞) is the diffusion constant, Δ 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 2d κ, split into two at rate ξ ∨ 0, and die at rate (−ξ) ∨ 0. In earlier work we looked at the Lyapunov exponents $ \lambda _{p}(\kappa ) = \lim\limits _{t\to \infty } \frac {1}{t} \log \mathbb {E} ([u(0,t)]^{p})^{1/p}, \quad p \in \mathbb{N} , \qquad \lambda _{0}(\kappa ) = \lim\limits _{t\to \infty } \frac {1}{t}\log u(0,t). $ For 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 κΔ is replaced by Δ$^{𝓚}$, where 𝓚 = {𝓚(x, y) : x, y ∈ ℤ$_{ d }$, x ∼ y} is a collection of random conductances between neighbouring sites replacing the constant conductances κ in the homogeneous model. We show that the associated annealed Lyapunov exponents λ $_{ p }$(𝓚), p ∈ ℕ, are given by the formula $ \lambda _{p}(\mathcal{K} ) = \text{sup} \{\lambda _{p}(\kappa ) : \, \kappa \in \text{Supp} (\mathcal{K} )\}, $ where, for a fixed realisation of 𝓚, Supp(𝓚) is the set of values taken by the 𝓚-field. 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.
Ending Page	36
Page Count	36
Starting Page	1
File Format	PDF
ISSN	13850172
e-ISSN	15729656
Journal	Mathematical Physics, Analysis and Geometry
Issue Number	2
Volume Number	19
Language	English
Publisher	Springer Netherlands
Publisher Date	2016-04-25
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Confinement Theoretical, Mathematical and Computational Physics Random conductances Dynamics of disordered systems (random Ising systems, etc.) Interacting random processes; statistical mechanics type models; percolation theory Large deviations Asymptotic behavior of solutions Geometry Random operators and equations Parabolic Anderson equation Lyapunov exponents Analysis Variational representations Group Theory and Generalizations Applications of Mathematics
Content Type	Text
Resource Type	Article
Subject	Mathematical Physics Geometry and Topology

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