Harnack Inequality and Gradient Estimate for $G$-SDEs with Degenerate Noise

In this paper, the Harnack inequalities for $G$-SDEs with degenerate noise are derived by method of coupling by change of measure. Moreover, the gradient estimate for the associated nonlinear semigroup $\bar{P}_t$ $$|\nabla \bar{P}_t f|\leq c(p,t)(\bar{P}_t |f|^p)^{\frac{1}{p}}, \ \ p>1, t>0$$ is also obtained for bounded and continuous function $f$. As an application of Harnack inequality, we prove the weak existence of degenerate $G$-SDEs under some integrable conditions. Finally, an example is presented. All of the above results extends the existed results in the linear expectation setting.