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

Abstract:

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.

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Received: 1 June 2001 / Accepted: 11 January 2002

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Ben Arous, G., Owhadi, H. Super-Diffusivity in a Shear Flow Model¶from Perpetual Homogenization. Commun. Math. Phys. 227, 281–302 (2002). https://doi.org/10.1007/s002200200640

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Keywords

  • Differential Equation
  • Spatial Scale
  • Asymptotic Behavior
  • Smooth Function
  • Flow Model