We prove the uniqueness of the martingale problem associated to some degenerate operators. The key point is to exploit the strong parallel between the new technique introduced by Bass and Perkins [BP09] to prove uniqueness of the martingale problem in the framework of non- degenerate elliptic operators and the Mc Kean and Singer [MS67] parametrix approach to the density expansion that has previously been extended to the degenerate setting that we consider (see Delarue and Menozzi [DM10]).
"Parametrix techniques and martingale problems for some degenerate Kolmogorov equations." Electron. Commun. Probab. 16 234 - 250, 2011. https://doi.org/10.1214/ECP.v16-1619