Ito-Wentzell-Lions formula for measure dependent random fields under full and conditional measure flows

TL;DR

This paper develops Ito-Wentzell formulae for measure-dependent random fields on Wiener spaces, addressing both full and marginal measure flows, with derivatives defined in Lions' sense, advancing mathematical tools for mean-field game analysis.

Contribution

It introduces new Ito-Wentzell formulae for measure-dependent fields, distinguishing between full and marginal measure flows, with derivatives in Lions' sense, enhancing stochastic analysis in mean-field contexts.

Findings

Derived Ito-Wentzell formulae for measure-dependent fields

Clarified derivatives with respect to measure components in Lions' sense

Applied results to mean-field game frameworks

Abstract

We present several Ito-Wentzell formulae on Wiener spaces for real-valued functional random field of Ito type that depend on measure flows. We distinguish the full- and the marginal-measure flow cases in the spirit of mean-field games. Derivatives with respect to the measure components are understood in the sense of Lions.