Geometric Ergodicity of Some Hypo-Elliptic Diffusions for Particle Motions
2002, v.8, Issue 2, 199-214
Two degenerate SDEs arising in statistical physics are studied. The first is a Langevin equation with state-dependent noise and damping. The second is the equation of motion for a particle obeying Stokes' law in a Gaussian random field; this field is chosen to mimic certain features of turbulence. Both equations are hypo-elliptic and smoothness of probability densities may be established. By developing appropriate Lyapunov functions and by studying the necessary control problems, geometric ergodicity is proved.
Keywords: geometric ergodicity,stochastic differential equations,Langevin equation,synthetic turbulence,hypoelliptic and degenerate diffusions