Ergodicity and Reversibility of the Minimal Diffusion Process

Fuxi Zhang

2005, v.11, №4, 661-676

ABSTRACT

In this paper, Prigogine's non-equilibrium steady state is modelled by a minimal diffusion process with only smoothness condition on the coefficients. We show that for such a process, the time reversal process is still a minimal diffusion process. We define an entropy production rate and provide three equivalent conditions of the reversibility of the process. Thus the existence of a reversible probability measure may be checked directly from the coefficients. For this paper to be self-contained, we first recall the construction of a minimal diffusion process whose Markov semigroup acts on a separable Banach space, which could not be the space of functions vanishing at infinity $C_0 (R^d)$. Since the coefficients may increase rapidly, $C_0 (R^d)$ is not invariant under the Markov semigroup. For this process, we show the Foguel alternative, i.e. the ergodicity of the Markov semigroup. Moreover, the ergodic theorem in the path space of a stationary process holds.

Keywords: diffusion process,ergodicity,irreversibility,entropy production rate