The Lions Derivative in Infinite Dimensions and Higher Order Expansion of Mean-Field SPDEs

Alexander Vogler, Wilhelm Stannat

Published 2024-07-20Version 1

In this paper we present a new interpretation of the Lions derivative as the Radon-Nikodym derivative of a vector measure, which provides a canonical extension of the Lions derivative for functions taking values in infinite dimensional Hilbert spaces. This is of particular relevance for the analysis of Hilbert space valued Mean-Field equations. As an illustration we derive a mild Ito-formula for Mean-Field SPDEs, which provides the basis for a higher order Taylor expansion and higher order numerical schemes.