Ergodicity of Langevin Processes with Degenerate Diffusion in Momentums

Details Ergodicity of Langevin Processes with Degenerate Diffusion in Momentums Ergodicity of Langevin Processes with Degenerate Diffusion in Momentums

2007-10-23

arxiv.org/abs/0710.4259v4

Mathematics

Probability

15 pages, to appear in International Journal of Pure and Applied Mathematics

Scientific paper

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

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