  The total entropy production of stochastic systems can be divided into three
quantities. The first corresponds to the excess heat, whilst the second two
comprise the house-keeping heat. We denote these two components the transient
and generalised house-keeping 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.
