Degenerate McKean--Vlasov SDE with Yamada--Osgood Coefficients and Arbitrary Modulus of Continuity in Wasserstein Distance

We establish the well-posedness of strong solutions for a class of degenerate mean-field stochastic differential equations with coefficients satisfying non-Lipschitz conditions. The diffusion coefficient is not required to be non-degenerate. Both the diffusion and the drift coefficients may depend on the law through the Kantorovich-Rubinstein metric with an arbitrary modulus of continuity. The drift and the diffusion satisfy a dimension-dependent Osgood-type condition, that becomes particularly sharp in dimension one. The proof combines the classical Yamada--Watanabe argument with Schauder's fixed-point theorem and does not rely on Gaussian estimates, allowing us to cover degenerate diffusions where such estimates are not available. This fills a gap in the existing literature.