It{\^o}-Krylov's formula for a flow of measures

Abstract

We prove It{\^o}'s formula for the flow of measures associated with an It{\^o} 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 It{\^o}-Krylov formula for functions in a Sobolev space on $\mathbf{R}^+ \times \mathbf{R}^d $.