Abstract

We consider the SDE degenerated at the boundary of a non-smooth domain. We prove uniqueness and existence of the martingale problem related to this degenerate SDEs under suitable non-negativity and regularity conditions on the coefficients. Applying martingale problem theory of Stroock and Varadhan, we turn the uniqueness problem of the SDE to the well-posedness of a kind of degenerate PDE with Neumann boundary condition defined on $R_{+}^{n}$. The difficulties for solvability of the problem mainly come from the degeneration of the operator, domain with corner, and correlation of different components of the SDE. The Schauder estimate for the degenerate PDE is given. We first estimate the mixed second order derivatives, and then utilize the perturbation method and the Schauder estimation for diagonal form to deal with the second order derivatives of normal direction on the corner boundary. This is a joint work with Kai Du.