Ergodicity of Langevin Processes with Degenerate Diffusion in Momentums

摘要

This paper introduces a geometric method for proving ergodicity of degeneratenoise driven stochastic processes. The driving noise is assumed to be anarbitrary Levy process with non-degenerate diffusion component (but that may beapplied to a single degree of freedom of the system). The geometric conditionsare the approximate controllability of the process the fact that there exists apoint in the phase space where the interior of the image of a point via asecondarily randomized version of the driving noise is non void. The paperapplies the method to prove ergodicity of a sliding disk governed byLangevin-type equations (a simple stochastic rigid body system). The papershows that a key feature of this Langevin process is that even though thediffusion and drift matrices associated to the momentums are degenerate, thesystem is still at uniform temperature.