Non-uniqueness for McKean--Vlasov SDEs with Hölder diffusion coefficients

We show that uniqueness may fail for one-dimensional McKean–Vlasov stochastic differential equations with Hölder continuous diffusion coefficients. More precisely, for every $\alpha \in (1\mathrm{/}2,1)$, we construct an equation with a diffusion coefficient that is $\alpha$-Hölder continuous in the state variable, Lipschitz continuous in the measure variable, and of linear growth, for which uniqueness in law and pathwise uniqueness fail. The construction relies on a spreading estimate together with a Schauder fixed-point argument.