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

Consider diffusions on R-k, k > 1, governed by the It (o) over cap equation dX(t) = {b(X(t)) + beta(X(t)/a)} dt + sigma dB(t), where b, beta are perio dic with the same period and are divergence free, cr is nonsingular and a i s a large integer. Two distinct Gaussian phases occur as time progresses. T he initial phase is exhibited over times 1 much less than t much less than a(2/3). Under a geometric condition on the velocity field beta, the final G aussian phase occurs for times t much greater than a(2)(log a)(2), and the dispersion grows quadratically with a. Under a complementary condition, the final phase shows up at times t much greater than a(4)(log a)(2), or t muc h greater than a(2) log a under additional conditions, with no unbounded gr owth in dispersion as a function of scale. Examples show the existence of n on-Gaussian intermediate phases. These probabilisitic results are applied t o analyze a multiscale Fokker-Planck equation governing solute transport in periodic porous media. In case b, beta are not divergence free, some insig ht is provided by the analysis of one-dimensional multiscale diffusions wit h periodic coefficients.