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

TL;DR

This paper establishes an Itô's formula for measure flows linked to Itô processes with bounded drift and diffusion, extending classical results to measure-dependent functions in Sobolev spaces.

Contribution

It introduces an Itô's formula for measure flows associated with bounded drift and elliptic diffusion, generalizing the Krylov formula to measure-dependent Sobolev functions.

Findings

Proves Itô's formula for measure flows with bounded coefficients

Extends Krylov's formula to measure-dependent Sobolev spaces

Provides a theoretical foundation for stochastic analysis on measure spaces

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 $.