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

Alexander Vogler; Wilhelm Stannat

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arXiv · 2407.14884

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

Alexander Vogler·Wilhelm Stannat

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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.

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BibTeX RIS

Alexander Vogler, Wilhelm Stannat. 2024-07-20. The Lions Derivative in Infinite Dimensions -- Application to Higher Order Expansion of Mean-Field SPDEs. https://arxiv.org/abs/2407.14884

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