The Green-Kubo formula and power spectrum of reversible Markov processes. (English) Zbl 1062.82042
Summary: 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.
MSC:
| 82C31 | Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics |
| 60J25 | Continuous-time Markov processes on general state spaces |
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