Fluctuation theorems in general stochastic processes with odd-parity variables

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.