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

Abstract: In this paper, we establish the It^o-Wentzell-Lions formulae for flows of both full and conditional measures on general semimartingales. This generalizes the existing works on flows of measures on It^o 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 It^o-Wentzell-Lions formulae for time-space-measure-dependent functions and for functions driven by Poisson random measures.