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-homogeneous diffusion processes whose spatially periodic dynamics, although time 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 are 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 ≪ t ≪ a2/3. Depending on the specifics of the dynamics, the final phase may show up reasonably fast, namely, for t ≫ a2 log a; or, it may take an enormous amount of time t ≫ exp{ca} for some c>0. An estimation of the speed of convergence to equilibrium of diffusions on a circle of circumference ‘a’ is provided for the above analysis.