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 = ∑_k=0ⁿ U _k(x/R_k) obtained as a superposition of Hölder-continuous periodic potentials U_k (of period double-struck T^d:= ℝ^d/ℤ^d, d ∈ N*, U_k (0) = 0) decays exponentially fast with the number of scales when the scale ratios R_k+1/R_k 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, dy_t = dω_t - ∇V^∞ (y_t)dt.