Itô-Wentzell-Lions Formula for Measure Dependent Random Fields under Full and Conditional Measure Flows

Abstract

We present several Itô-Wentzell formulae on Wiener spaces for real-valued functional random field of Itô 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.