Speed of convergence to equilibrium and to normality for diffusions with multiple periodic scales

The present article analyses the large-time behavior of a class of time-hom ogeneous diffusion processes whose spatially periodic dynamics, although ti me independent, involve a large spatial parameter 'a'. This leads to phase changes in the behavior of the process as time increases through different time zones. At least four different temporal regimes can be identified: an initial non-Gaussian phase for times which an not large followed by a first Gaussian phase, which breaks down over a subsequent region of time, and a final Gaussian phase different from the earlier phases. The first Gaussian phase occurs for times 1 much less than t much less than a (2/3). Depending on the specifics of the dynamics, the final phase may show up reasonably f ast, namely, for t much greater than a(2) log a; or, ii may take an enormou s amount of time t much greater than exp{ca} for some c>0. An estimation of the speed of convergence to equilibrium of diffusions on a circle of circu mference 'a' is provided for the above analysis. (C) 1999 Elsevier Science B.V. All rights reserved.