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We investigate the decomposition of the total entropy production incontinuous stochastic dynamics when there are odd-parity variables that changetheir signs under time reversal. The first component of the entropy production,which satisfies the fluctuation theorem, is associated with the usual excessheat that appears during transitions between stationary states. The remaininghousekeeping part of the entropy production can be further split into twoparts. We show that this decomposition can be achieved in infinitely many wayscharacterized by a single parameter {\sigma}. For an arbitrary value of{\sigma}, one of the two parts contributing to the housekeeping entropyproduction satisfies the fluctuation theorem. We show that for a range of{\sigma} values this part can be associated with the breakage of the detailedbalance in the steady state, and can be regarded as a continuous version of thecorresponding entropy production that has been obtained previously for discretestate variables. The other part of the housekeeping entropy does not satisfythe fluctuation theorem and is related to the parity asymmetry of thestationary state distribution. We discuss our results in connection with thedifference between continuous and discrete variable cases especially in theconditions for the detailed balance and the parity symmetry of the stationarystate distribution.

Housekeeping Entropy in Continuous Stochastic Dynamics with Odd-Parity Variables