  We show that the total entropy production in stochastic processes with
odd-parity variables (under time reversal) is separated into three parts, only
two of which satisfy the integral fluctuation theorems in general. One is the
usual excess entropy production, which can appear only transiently and is
called nonadiabatic. Another one is attributed solely to the breakage of
detailed balance. The last part not satisfying the fluctuation theorem comes
from the steady-state distribution asymmetry for odd-parity variables, which is
activated in a non-transient manner. The latter two contributions combine
together as the house-keeping (adiabatic) entropy production, whose positivity
is not guaranteed except when the excess entropy production completely
vanishes.
