Generalization of Taylor's formula to particles of arbitrary inertia

One of the cornerstones of turbulent dispersion is the celebrated Taylor formula. This formula expresses the rate of transport (i.e. the eddy diffusivity) of a tracer as a time integral of the fluid velocity auto-correlation function evaluated along the fluid trajectories. Here, we review the hypotheses which permit to extend Taylor's formula to particles of any inertia. The hypotheses are independent of the details of the inertial particle model.We also show by explicit calculation, that the hypotheses encompass cases when memory terms such as Basset and the Fax\'en corrections are taken into account in the modeling of inertial particle dynamics.