The Onsager-Machlup action functional for Mckean-Vlasov SDEs

TL;DR

This paper derives the Onsager-Machlup action functional for McKean-Vlasov SDEs within various normed spaces, providing the associated Euler-Lagrange equations and an illustrative example.

Contribution

It introduces the Onsager-Machlup functional for McKean-Vlasov SDEs in a broad class of norms, extending previous formulations and deriving the related Euler-Lagrange equations.

Findings

Derived Onsager-Machlup functional for McKean-Vlasov SDEs in multiple norms.

Established the Euler-Lagrange equation for the functional.

Provided a concrete example illustrating the theory.

Abstract

This paper is devoted to deriving the Onsager-Machlup action functional for Mckean-Vlasov stochastic differential equations in a class of norms that dominate $L^2([0,1], \mathbb{R}^d)$, such as supremum norm $\|\cdot\|_{\infty}$, H$\mathrm{\ddot{o}}$lder norms $\|\cdot\|_{\alpha}$ with $\alpha<\frac{1}{4}$ and $L^p$-norms with $p>4$ are included. Moreover, the corresponding Euler-Lagrange equation for Onsager-Machlup action functional is derived and a example is given.