  We investigate the decomposition of the total entropy production in
continuous stochastic dynamics when there are odd-parity variables that change
their signs under time reversal. The first component of the entropy production,
which satisfies the fluctuation theorem, is associated with the usual excess
heat that appears during transitions between stationary states. The remaining
housekeeping part of the entropy production can be further split into two
parts. We show that this decomposition can be achieved in infinitely many ways
characterized by a single parameter {\sigma}. For an arbitrary value of
{\sigma}, one of the two parts contributing to the housekeeping entropy
production satisfies the fluctuation theorem. We show that for a range of
{\sigma} values this part can be associated with the breakage of the detailed
balance in the steady state, and can be regarded as a continuous version of the
corresponding entropy production that has been obtained previously for discrete
state variables. The other part of the housekeeping entropy does not satisfy
the fluctuation theorem and is related to the parity asymmetry of the
stationary state distribution. We discuss our results in connection with the
difference between continuous and discrete variable cases especially in the
conditions for the detailed balance and the parity symmetry of the stationary
state distribution.
