This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dyt = dωt - ∇(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∞ 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 (double-struck E sign [|ytt|2] ∼ t 1+v with v > 0) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization.
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics