  Stochastic entropy production, which quantifies the difference between the
probabilities of trajectories of a stochastic dynamics and its time reversals,
has a central role in nonequilibrium thermodynamics. In the theory of
probability, the change in the statistical properties of observables can be
represented by a change in the probability measure. We consider operators on
the space of probability measure that induce changes in the statistical
properties of a process, and formulate entropy productions in terms of these
change-of-probability-measure (CPM) operators. This mathematical underpinning
of the origin of entropy productions allows us to achieve an organization of
various forms of fluctuation relations: All entropy productions have a
non-negative mean value, admit the integral fluctuation theorem, and satisfy a
rather general fluctuation relation. Other results such as the transient
fluctuation theorem and detailed fluctuation theorems then are derived from the
general fluctuation relation with more constraints on the operator. We use a
discrete-time, discrete-state-space Markov process to draw the
contradistinction among three reversals of a process: time reversal, protocol
reversal and the dual process. The properties of their corresponding CPM
operators are examined, and the domains of validity of various fluctuation
relations for entropy productions in physics and chemistry are revealed. We
also show that our CPM operator formalism can help us rather easily extend
other fluctuations relations for excess work and heat, discuss the martingale
properties of entropy productions, and derive the stochastic integral formulas
for entropy productions in constant-noise diffusion process with Girsanov
theorem. Our formalism provides a general and concise way to study the
properties of entropy-related quantities in stochastic thermodynamics and
information theory.
