Multiscale Homogenization with Bounded Ratios and Anomalous Slow Diffusion

Gérard Ben Arous, Houman Owhadi

Research output: Contribution to journalArticle


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/Rk) obtained as a superposition of Hölder-continuous periodic potentials Uk (of period double-struck 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.

Original languageEnglish (US)
Pages (from-to)80-113
Number of pages34
JournalCommunications on Pure and Applied Mathematics
Issue number1
StatePublished - Jan 1 2003


ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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