Weak uniqueness is established for the martingale problem associated to a family of catalytic branching networks. This martingale problem corresponds to a stochastic differential equation with a degenerate Hölder continuous diffusion matrix. Our approach uses the semigroup perturbation method of Stroock and Varadhan and a modification of a Banach space of weighted Hölder continuous functions introduced by Bass and Perkins.
"On the uniqueness problem for catalytic branching networks and other singular diffusions." Illinois J. Math. 50 (1-4) 323 - 383, 2006. https://doi.org/10.1215/ijm/1258059478