The entropy production and circulation of diffusion processes on manifolds

摘要

IT is known that the irreversibility of a Markov chain can be characterized by its circulation (see e.g. ref.). In this note, an extension of Qian-Qian's result on the circulation for Markov chains to the case of diffusion processes on manifolds is presented. It is shown that the entropy production of a diffusion {x_t}_t0 on a manifold M is represented as a linear sum of the rotation numbers (circulation) of {x_t}_t0 around some closed curves in M and the circulation α_0 of a lifting process onM X S~1 around the circle S~1.