The Onsager-Machlup action functional for Mckean-Vlasov SDEs

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 $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{\infty }}$, H$\ddot{\mathrm{o}}$lder norms $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\alpha }$ 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.