  The entropy production rate is a central quantity in non-equilibrium
statistical physics, scoring how far a stochastic process is from being
time-reversible. In this paper, we compute the entropy production of diffusion
processes at non-equilibrium steady-state under the condition that the
time-reversal of the diffusion remains a diffusion. We start by characterising
the entropy production of both discrete and continuous-time Markov processes.
We investigate the time-reversal of time-homogeneous stationary diffusions and
recall the most general conditions for the reversibility of the diffusion
property, which includes hypoelliptic and degenerate diffusions, and locally
Lipschitz vector fields. We decompose the drift into its time-reversible and
irreversible parts, or equivalently, the generator into symmetric and
antisymmetric operators. We show the equivalence with a decomposition of the
backward Kolmogorov equation considered in hypocoercivity theory, and a
decomposition of the Fokker-Planck equation in GENERIC form. The main result
shows that when the time-irreversible part of the drift is in the range of the
volatility matrix (almost everywhere) the forward and time-reversed path space
measures of the process are mutually equivalent, and evaluates the entropy
production. When this does not hold, the measures are mutually singular and the
entropy production is infinite. We verify these results using exact numerical
simulations of linear diffusions. We illustrate the discrepancy between the
entropy production of non-linear diffusions and their numerical simulations in
several examples and illustrate how the entropy production can be used for
accurate numerical simulation. Finally, we discuss the relationship between
time-irreversibility and sampling efficiency, and how we can modify the
definition of entropy production to score how far a process is from being
generalised reversible.
