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

Alexander Vogler, Wilhelm Stannat

TL;DR

The paper advances the theory of Lions derivatives by recasting the L-differentiability for Hilbert-space-valued maps as a Radon–Nikodym derivative of a vector measure, yielding a density $\frac{dm_{\mu}}{d\mu}$ that depends only on $\mu=\mathcal{L}(X)$. This intrinsic, lift-independent representation enables a mild Itô calculus for flows of measures in infinite dimensions and, consequently, higher-order Taylor expansions for Mean-Field SPDEs. The authors establish a general framework, proving a main theorem that provides the regular L-derivative $\partial_\mu f(\mu)$ and disintegration-based constructions, alongside detailed discrete and general-case analyses and illustrative examples. They then develop a mild Itô formula for MFSPDEs, derive stochastic Taylor expansions up to order $2+\min(\delta,\gamma)$, and discuss applications to control and maximum-principle formulations, highlighting the practical impact on numerical schemes and optimal control of mean-field systems in infinite dimensions.

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.

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

TL;DR

The paper advances the theory of Lions derivatives by recasting the L-differentiability for Hilbert-space-valued maps as a Radon–Nikodym derivative of a vector measure, yielding a density that depends only on . This intrinsic, lift-independent representation enables a mild Itô calculus for flows of measures in infinite dimensions and, consequently, higher-order Taylor expansions for Mean-Field SPDEs. The authors establish a general framework, proving a main theorem that provides the regular L-derivative and disintegration-based constructions, alongside detailed discrete and general-case analyses and illustrative examples. They then develop a mild Itô formula for MFSPDEs, derive stochastic Taylor expansions up to order , and discuss applications to control and maximum-principle formulations, highlighting the practical impact on numerical schemes and optimal control of mean-field systems in infinite dimensions.

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.
Paper Structure (19 sections, 30 theorems, 288 equations)

This paper contains 19 sections, 30 theorems, 288 equations.

Table of Contents

  1. Introduction and Main Result
  2. Preliminaries
  3. Notation
  4. Vector Measures
  5. Integration w.r.t. Vector Measures of Finite Variation
  6. Vector Measures of Finite (Semi) q-Variation
  7. Linear Operators on $\mathcal{L}^p$
  8. Generalized L-Derivative, Proof of Theorem \ref{['g']}
  9. The Discrete Case
  10. The General Case
  11. Structure of the L-Derivative
  12. Examples
  13. Higher Order Expansion for Mean-Field SPDEs
  14. Itô-Formula for Flow of Measures in Infinite Dimensions
  15. A Mild Itô-Formula for Mean-Field SPDEs
  16. ...and 4 more sections

Key Result

Lemma 1.2

Let $H$ be a Hilbert space, $U$ be a Banach space and $\hat{f} : L^2 (\Omega , \mathcal{F}, \mathbb{P}; H) \rightarrow U$ be Fréchet differentiable in $X_0\in L^2(\Omega, \mathcal{F}, \mathbb{P};H)$, then defines a vector measure, see Definition DefVecMeas,

Theorems & Definitions (68)

  • Definition 1.1
  • Lemma 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 58 more