Discontinuous superprocesses with dependent spatial motion

We construct a class of discontinuous superprocesses with dependent spatial motion and general branching mechanism. The process arises as the weak limit of critical interacting–branching particle systems where the spatial motions of the particles are not independent. The main work is to solve the martingale problem. When we turn to the uniqueness of the process, we generalize the localization method introduced by [Daniel W. Stroock, Diffusion processes associated with Lévy generators, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975) 209–244] to the measure-valued context. As for existence, we use particle system approximation and a perturbation method. This work generalizes the model introduced in [Donald A. Dawson, Zenghu Li, Hao Wang, Superprocesses with dependent spatial motion and general branching densities, Electron. J. Probab. 6 (25) (2001) 33 pp (electronic)] where a quadratic branching mechanism was considered. We also investigate some properties of the process.