In this paper, we first find a type of viscosity solution of $G$-heat equation under degenerate case, and then obtain the related $G$-capacity $c({B_{T}in A})$ for any Borel set $A$. Furthermore, we prove that $I_{A}(B_{T})$ is not quasi-continuous when it is not a constant function.
In this paper, we study a stochastic recursive optimal control problem in which the value functional is defined by the solution of a backward stochastic differential equation (BSDE) under $tilde{G}$-expectation. Under standard assumptions, we establi
sh the comparison theorem for this kind of BSDE and give a novel and simple method to obtain the dynamic programming principle. Finally, we prove that the value function is the unique viscosity solution of a type of fully nonlinear HJB equation.
