Harnack Inequality for Distribution Dependent Stochastic Hamiltonian System

Abstract

By investigating the regularity of the nonlinear semigroup P * t associated to the distribution dependent stochastic Hamiltonian system, the Harnack inequality is derived when the drift is Lipschitz continuous in the measure variable under the distance induced by the functions which are β(β > 23 )-Hölder continuous on the degenerate component and square root of Dini continuous on the non-degenerate one. The results extend the ones in the case where the drifts are Lipschitz continuous in L 2 -Wasserstein distance.