### Abstract

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations dy_{t} = dω_{t} - ∇(y_{t})dt, y_{0} = 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 Γ_{12} = - Γ_{21} = 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 (double-struck E sign [|y_{t}t|^{2}] ∼ t ^{1+v} with v > 0) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization.

Original language | English (US) |
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Pages (from-to) | 281-302 |

Number of pages | 22 |

Journal | Communications In Mathematical Physics |

Volume | 227 |

Issue number | 2 |

DOIs | |

State | Published - May 1 2002 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications In Mathematical Physics*,

*227*(2), 281-302. https://doi.org/10.1007/s002200200640