Onsager--Machlup functionals for McKean--Vlasov SDEs via Euler-type approximation

The Onsager–Machlup action functional provides a variational framework for characterizing the most probable transition paths of stochastic systems and plays an important role in the study of nonequilibrium fluctuations. Its extension to McKean–Vlasov stochastic differential equations is complicated by the intrinsic distribution dependence of the coefficients. In this paper, we address this difficulty by introducing an Euler-type approximation scheme based on classical, distribution-free stochastic differential equations. Combining the classical Onsager–Machlup theory with a convergence argument for the approximation sequence, we derive an explicit expression for the Onsager–Machlup functional associated with the McKean–Vlasov SDE. The proposed approach is constructive and extends to a broad class of distribution-dependent stochastic systems.