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