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

The solution of ihe Itô equation dX(t) = b{X(t)}dt + β{X(t)/a)dt + √DdB(t) is analysed for t → ∞, a → ∞. In the range 1 < t < a^2/3, X(t) is asymptotically Gaussian if b is periodic, β Lipschitzian; here the large-scale fluctuations may be ignored. In the range t > a², with both b and β periodic and divergence-free, a integral, Gaussian approximation is again valid under an appropriate hypothesis on the geometry of β here for some coordinates of X(t) the dispersivity. or variance per unit time, mav grow at the extreme rate O(a²) white stabilizing for others. As shown by examples, Gaussian approximation generally breaks down at intermediate lime-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.