Speed of convergence to equilibrium and to normality for diffusions with multiple periodic scales
Let \(X\) be a scalar Itô diffusion governed by an equation of the form \[ X(t)= x_0 + \int _0^t\{b(X(s))+\beta (X(s)/a)\} ds +\int _0^t \sigma (X(s)) dB(s), \] where \(b\) is thought of as the local drift velocity and \(\beta (x/a)\) the large-scale velocity of \(X\), the spatial scale parameter \(a\) being large. Behaviour of the diffusion exhibits phase changes as time increases through the different time zones. Four different temporal regimes are identified: an initial non-Gaussian phase for times that 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 is shown to occur for times \(1\ll t\ll a^{2/3}\). Detailed analysis of the final phase is carried out as well.
scientific article; zbMATH DE number 1554292
scientific article