Multiscale diffusion processes with periodic coefficients and an application to solute transport in porous media

Consider diffusions on Rk>1, governed by the Itô equation dX(t)=b(X(t))+.(X(t)/a)dt+\sigmadB(t), where b,. are periodic with the same period and are divergence free, . is nonsingular and a is a large integer. Two distinct Gaussian phases occur as time progresses. The initial phase is exhibited over times 1.t.a2/3. Under a geometric condition on the velocity field ., the final Gaussian phase occurs for times t.a2(loga)2, and the dispersion grows quadratically with a . Under a complementary condition, the final phase shows up at times t.a4(loga)2, or t.a2loga under additional conditions, with no unbounded growth in dispersion as a function of scale. Examples show the existence of non-Gaussian intermediate phases. These probabilisitic results are applied to analyze a multiscale Fokker-Planck equation governing solute transport in periodic porous media. In case b,. are not divergence free, some insight is provided by the analysis of one-dimensional multiscale diffusions with periodic coefficients.