Itô-Wentzell-Lions formulae for flows of full and conditional measures on semimartingales

In this paper, we establish the Itô-Wentzell-Lions formulae for flows of both full and conditional measures on general semimartingales. This generalizes the existing works on flows of measures on Itô processes. The key technical components involve an appropriate approximation of random fields by cylindrical functions and localization techniques. Moreover, we present the specific formulae in two special cases, including Itô-Wentzell-Lions formulae for time-space-measure-dependent functions and for functions driven by Poisson random measures.