The quasiderivative method for derivative estimates of solutions to degenerate elliptic equations

We give an example of quasiderivatives constructed by random time change, Girsanov’s Theorem and Levy’s Theorem. As an application, we investigate the smoothness and estimate the derivatives up to second order for the probabilistic solution to the Dirichlet problem for the linear degenerate elliptic partial differential equation of second order, under the assumption of non-degeneracy with respect to the normal to the boundary and an interior condition to control the moments of quasiderivatives, which is weaker than non-degeneracy.