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We present a Lyapunov type approach to the problem of existence anduniqueness of general law-dependent stochastic differential equations. In theexisting literature most results concerning existence and uniqueness areobtained under regularity assumptions of the coefficients w.r.t the Wassersteindistance. Some existence and uniqueness results for irregular coefficients havebeen obtained by considering the total variation distance. Here we extend thisapproach to the control of the solution in some weighted total variationdistance, that allows us now to derive a rather general weak uniqueness result,merely assuming measurability and certain integrability on the driftcoefficient and some non-degeneracy on the dispersion coefficient. We alsopresent an abstract weak existence result for the solution of law-dependentstochastic differential equations with merely measurable coefficients, based onan approximation with law-dependent stochastic differential equations withregular coefficients under Lyapunov type assumptions.

Weak Solutions to Vlasov-McKean Equations under Lyapunov-Type Conditions