The paper considers general diffusion processes taking place on widely
separated time scales, and presents a measure-theoretic approach to analyze
various general thermodynamic functionals along sample paths, which satisfy
fluctuation theorems. We find out that the limit of these functionals with odd
and even variables under first-order and second-order singular perturbations
fails to be the one directly defined on the limiting diffusion processes after
the rapid dimensions have been eliminated. Their difference is called an
anomalous term, which turns out to be exponential martingales in all cases
under consideration and satisfies the fluctuation theorems. Sufficient and
necessary conditions for the vanishing of these anomalous terms are rigorously
derived. Physical applications have also been included, especially for the
second-order diffusion processes towards overdamping limit. It is found that
the anomalous contribution of total entropy production can only emerge from the
averaged housekeeping heat, while the dissipative work in Jarzynski equality
can never contain the anomalous contribution.