"Derivation of the trace formula for a deterministic flow perturbed by
weak additive force"

looks like a honest derivation requires step-by-step repeat of what we
are doing for QM; path integrals, jacobians galore, Onsager-Machlup
variational principle, Ito vs Stratanovich (this is called "operator
ordering problem" in QM), etc.

No numerical work, but quite bit of reading, thinking
and careful analytic work.