It{ô}-Krylov's formula for a flow of measures

Abstract

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