Communications in Mathematical Physics

, Volume 227, Issue 2, pp 281-302

First online:

Super-Diffusivity in a Shear Flow Model¶from Perpetual Homogenization

  • Gérard Ben ArousAffiliated withDMA, EPFL, 1015 Lausanne, Switzerland. E-mail:
  • , Houman OwhadiAffiliated withWilliam Davidson Faculty (Bloomfield), Technion, 32000 Haifa, Israel. E-mail:

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This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dy t =dω t −∇Γ(y t ) dt, y 0=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(x 1), with h(x 1)=∑ n =0 γ n h n (x 1/R n ), where h n are smooth functions of period 1, h n (0)=0, γ n and R n grow exponentially fast with n. We can show that y t has an anomalous fast behavior (?[|y t |2]∼t 1+ν with ν > 0) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization.