Several perturbation tools are established in the volume-preserving setting allowing for the pasting, extension, localized smoothing and local linearization of vector fields. The pasting and the local linearization hold in all classes of regularity ranging from $C^{1}$ to $C^{\infty }$ (Hölder included). For diffeomorphisms, a conservative linearized version of Franks’ lemma is proved in the $C^{r,\unicode[STIX]{x1D6FC}}$ ($r\in \mathbb{Z}^{+}$, $0<\unicode[STIX]{x1D6FC}<1$) and $C^{\infty }$ settings, the resulting diffeomorphism having the same regularity as the original one.

Weak local linear approximations have played a prominent role in the construction of effective inference methods and numerical integrators for stochastic differential equations. In this note two weak local linear approximations for stochastic differential equations with jumps are introduced as a generalization of previous ones. Their respective order of convergence is obtained as well.

Stochastic dynamical system models for non-linear random vibrations are considered and their discrete-time version, non-linear time series models are introduced using the local Gaussian modelling method. Some computational problems and implications of the present method in non-linear time series analysis are discussed.