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 &\times; 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∞γnhn(x1/Rn), where hn 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 (?[|yt|2]∼t1+ν with ν > 0) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization.
Unable to display preview. Download preview PDF.