Itô-Krylov’s Formula for a Flow of Measures

Abstract

We prove Itô's formula for the flow of measures associated with an Itô process having a bounded drift and a uniformly elliptic and bounded diffusion matrix, and for functions in an appropriate Sobolev-type space. This formula is the almost analogue, in the measure-dependent case, of the Itô-Krylov formula for functions in a Sobolev space on $\mathbb{R}^+ \times \mathbb{R}^d$.