On pathwise uniqueness of solutions for multidimensional McKean–Vlasov equation

Pathwise uniqueness for the multidimensional stochastic McKean–Vlasov equation is established under moderate regularity conditions on the drift and diffusion coefficients. Both drift and diffusion depend on the marginal measure of the solution. It is assumed that both coefficients are bounded, and, moreover, the drift is Dini-continuous in the state variable, and the diffusion satisfies the Lipschitz condition and is also continuous in time and uniformly nondegenerate. This is the classical McKean–Vlasov setting, that is, the coefficients of the equation are represented as integrals over the marginal distributions of the process.