Harnack Inequality for Distribution Dependent Second-Order Stochastic Differential Equations

By investigating the regularity of the nonlinear semigroup \(P_t^*\) $P_t^{\ast }$ 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 \(\beta \) $\beta$ -Hölder continuous (with \(\beta > \frac{2}{3}\) $\beta >\frac{2}{3}$ ) 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 \(L^2\) $L^2$ -Wasserstein distance.