The fluctuations of dynamical functionals such as the empirical density and
current as well as heat, work and generalized currents in stochastic
thermodynamics are usually studied within the Feynman-Kac tilting formalism,
which in the Physics literature is typically derived by some form of
Kramers-Moyal expansion, or in the Mathematical literature via the
Cameron-Martin-Girsanov approach. Here we derive the Feynman-Kac theory for
general additive dynamical functionals directly via It\^o calculus and via
functional calculus, where the latter result in fact appears to be new. Using
Dyson series we then independently recapitulate recent results on steady-state
(co)variances of general additive dynamical functionals derived recently in
Dieball and Godec ({2022 \textit{Phys. Rev. Lett.}~\textbf{129} 140601}) and
Dieball and Godec ({2022 \textit{Phys. Rev. Res.}~\textbf{4} 033243}). We hope
for our work to put the different approaches to the statistics of dynamical
functionals employed in the field on a common footing, and to illustrate more
easily accessible ways to the tilting formalism.
