  We analyze the stochastic thermodynamics of systems with continuous space of
states. The evolution equation, the rate of entropy production, and other
results are obtained by a continuous time limit of a discrete time formulation.
We point out the role of time reversal and of the dissipation part of the
probability current on the production of entropy. We show that the rate of
entropy production is a bilinear form in the components of the dissipation
probability current with coefficients being the components of the precision
matrix related to the Gaussian noise. We have also analyzed a type of noise
that makes the energy function to be strictly constant along the stochastic
trajectory, being appropriate to describe an isolated system. This type of
noise leads to nonzero entropy production and thus to an increase of entropy in
the system. This result contrasts with the invariance of the entropy predicted
by the Liouville equation, which also describes an isolated system.
