Time-scales for Gaussian approximation and its breakdown under a hierarchy of periodic spatial heterogeneities

Abstract

The solution of the Itô equation $dX\left(t\right)=b\left\{X\right(t\left)\right\}dt+\beta \left\{X\right(t)/a\}dt+\sqrt{bD}dB\left(t\right)$ is analysed for $t\to \mathrm{\infty }$, $a\to \mathrm{\infty }$. In the range $1\ll t\ll a^{2/3}$, $X\left(t\right)$ is asymptotically Gaussian if $b$ is periodic, $\beta$ Lipschitzian; here the large-scale fluctuations may be ignored. In the range $t\gg a^2$, with both $b$ and $\beta$ periodic and divergence-free, $a$ integral, Gaussian approximation is again valid under an appropriate hypothesis on the geometry of $\beta$; here for some coordinates of $X\left(t\right)$ the dispersivity, or variance per unit time, may grow at the extreme rate $O\left(a^2\right)$ while stabilizing for others. As shown by examples, Gaussian approximation generally breaks down at intermediate time-scales. These results translate into asymptotics of a class of Fokker-Planck equations which arise in the prediction of contaminant transport in an aquifer under multiple scales of spatial heterogeneity. In particular, contrary to popular belief, the growth in dispersivity is always slower than linear.