Super-diffusivity in a shear flow model from perpetual homogenization

Abstract

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations $dy_t=d\omega_t -\nabla \Gamma(y_t) dt$, $y_0=0$ and $d=2$. $\Gamma$ is a $2\times 2$ skew-symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales $\Gamma_{12}=-\Gamma_{21}=h(x_1)$, with $h(x_1)=\sum_{n=0}^\infty \gamma_n h^n(x_1/R_n)$ where $h^n$ are smooth functions of period 1, $h^n(0)=0$, $\gamma_n$ and $R_n$ grow exponentially fast with $n$. We can show that $y_t$ has an anomalous fast behavior ($\E[|y_t|^2]\sim t^{1+\nu}$ with $\nu>0$) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization.