The Green–Kubo formula and power spectrum of reversible Markov processes

As is known, the entropy production rate of a stationary Markov process vanishes if and only if the process is reversible. In this paper, we discuss the reversibility of a stationary Markov process from a functional analysis point of view. It is shown that the process is reversible if and only if it has a symmetric Markov semigroup, equivalently, a self-adjoint infinitesimal generator. Applying this fact, we prove that the Green–Kubo formula holds for reversible Markov processes. By demonstrating that the power spectrum of each reversible Markov process is Lorentz-typed, we show that it is impossible for stochastic resonance to occur in systems with zero entropy production.