Nonequilibrium thermodynamics of stochastic systems with odd and even variables

The total entropy production of stochastic systems can be divided into three quantities. The first corresponds to the excess heat, while the second two comprise the housekeeping heat. We denote these two components the transient and generalized housekeeping heat and we obtain an integral fluctuation theorem for the latter, valid for all Markovian stochastic dynamics. A previously reported formalism is obtained when the stationary probability distribution is symmetric for all variables that are odd under time reversal, which restricts consideration of directional variables such as velocity.