Multiscale homogenization with bounded ratios and anomalous slow diffusion



We show that the effective diffusivity matrix D(Vn) for the heat operator ∂t − (Δ/2 − ∇Vn∇) in a periodic potential Vn = Σmath imageUk(x/Rk) obtained as a superposition of Hölder-continuous periodic potentials Uk (of period ��d := 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.