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# Harnack Inequality for Distribution Dependent Stochastic Hamiltonian System

Aug 2022

By investigating the regularity of the nonlinear semigroup $P_t^\ast$associated to the distribution dependent stochastic Hamiltonian system, theHarnack inequality is derived when the drift is Lipschitz continuous in themeasure variable under the distance induced by the functions which are $\beta(\beta>\frac{2}{3})$-H\"{o}lder continuous on the degenerate component andsquare root of Dini continuous on the non-degenerate one. The results extendthe ones in the case where the drifts are Lipschitz continuous in$L^2$-Wasserstein distance.

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