Super-diffusivity in a shear flow model from perpetual homogenization

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dyt = dωt − ∇Γ(yt)dt, y0 = 0 and d = 2. Γ is a 2 × 2 skew-symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales Γ12 = −Γ21 = h(x1), with h(x1) = ∑ ∞ n=0 γnh (x1/Rn) where h n are smooth functions of period 1, hn(0) = 0, γn and Rn grow exponentially fast with n. We can show that yt has an anomalous fast behavior (E[|yt|] ∼ t1+ν with ν > 0) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization.