Harnack Inequality for Distribution Dependent Second-Order Stochastic Differential Equations

By investigating the regularity of the nonlinear semigroup \(P_t^*\) associated with the distribution dependent second-order stochastic differential equations, the Harnack inequality is derived when the drift is Lipschitz continuous in the measure variable under the distance induced by the functions being \(\beta \)-Hölder continuous (with \(\beta > \frac{2}{3}\)) on the degenerate component and square root of Dini continuous on the non-degenerate one. The results extend the existing ones in which the drift is Lipschitz continuous in \(L^2\)-Wasserstein distance.