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

TL;DR

This paper introduces a new interpretation of the Lions derivative as a Radon-Nikodym derivative in infinite dimensions, enabling advanced analysis and numerical schemes for Hilbert space valued Mean-Field SPDEs.

Contribution

It extends the Lions derivative to infinite-dimensional Banach spaces and applies this to derive a mild Ito-formula for higher order expansions of Mean-Field SPDEs.

Findings

New interpretation of Lions derivative as Radon-Nikodym derivative

Derivation of mild Ito-formula for Mean-Field SPDEs

Foundation for higher order Taylor expansions and numerical schemes

Abstract

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 Banach 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 stochastic partial differential equations (SPDEs), which provides the basis for a higher order Taylor expansion and higher order numerical schemes.