Nonlinear master equation of multitype particle systems

In this paper we prove the existence and uniqueness of solutions of the nonlinear martingale problems associated with the nonlinear master equations of multitype particle systems. Existence is shown to hold under some weak growth conditions and a finite range condition, while uniqueness is proved under a Lipschitz condition. Uniqueness is also shown to hold under a non-Lipschitz condition and a strong growth condition. The proof of uniqueness involves a coupling argument and the exponential martingale property. The results are then applied to some examples such as the Lotka-Volterra model and the Brusselator.