Multiscale homogenization with bounded ratios and anomalous slow diffusion
Gérard Ben Arous, Houman Owhadi · Communications on Pure and Applied Mathematics · 2002
Abstract We show that the effective diffusivity matrix D(Vn) for the heat operator ∂t − (Δ/2 − ∇Vn∇) in a periodic potential Vn = Σ Uk(x/Rk) obtained as a superposition of Hölder‐continuous periodic potentials Uk (of period 𝕋d := ℝd/ℤd, d ∈ ℕ*, Uk(0) = 0) decays exponentially fast with the number of scales when the scale ratios Rk+1/Rk are bounded above and below. From this we deduce the anomalous slow behavior for a Brownian motion in a potential obtained as a superposition of an infinite number of scales, dyt = dωt − ∇V∞(yt)dt. © 2002 Wiley Periodicals, Inc.