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 Itô equation d X (t)={b(X(t)) + β(X(t)/a)}dt + σdB(t), where b, β are periodic with the same period and are divergence free, σ 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 ≪ a^2/3. Under a geometric condition on the velocity field β, the final Gaussian phase occurs for times t ≫ a²(log a)², and the dispersion grows quadratically with a. Under a complementary condition, the final phase shows up at times t ≫ a⁴(log a)², or t ≫ a² log a 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, β are not divergence free, some insight is provided by the analysis of one-dimensional multiscale diffusions with periodic coefficients.