Regularities for distribution dependent SDEs with fractional noises

In this paper, we investigate the regularities for a class of distribution dependent SDEs driven by two independent fractional noises $B^H$ and $\text{ti}{B}^{\text{ti}H}$ with Hurst parameters $H\in (0,1)$ and $\text{ti}H\in (1\mathrm{/}2,1)$. We establish the log-Harnack inequalities and Bismut formulas for the Lions derivative to this type of equations with distribution dependent noise, in both non-degenerate and degenerate cases. Our proofs consist of utilizing coupling arguments which are indeed backward couplings introduced by F.-Y. Wang \cite{Wang12b}, together with a careful analysis of fractional derivative operator.