Equilibrium stochastic delay processes

Stochastic processes with temporal delay play an important role in science and engineering whenever finite speeds of signal transmission processing occur. However, exact mathematical analysis their dynamics thermodynamics is available for linear models only. We introduce a class stochastic nonlinear time-local forces time-delayed that obey fluctuation theorems converge to Boltzmann equilibrium at long times. From the point view control theory, such ``equilibrium processes'' are stable energetically passive, by construction. Computationally, they provide diverse constraints on general problems can, various situations, serve as starting perturbative analysis. Physically, admit interpretation terms underdamped Brownian particle either subjected force non-Markovian thermal bath or delayed feedback Markovian bath. illustrate these properties numerically setup familiar from cooling out experimental implications.