Finite range jump processes and volume–constrained diffusion problems

A nonlocal convection–diffusion model is introduced for the master equation of Markov jump processes in bounded domains. With minimal assumptions on the model parameters, the nonlocal steady and unsteady state master equations are shown to be well–posed in a weak sense. Then, the nonlocal operator is shown to be the generator of finite range nonsymmetric jump processes and, when certain conditions on the model parameters hold, the generators of finite and infinite activity Levy and Levy–type jump processes are shown to be special instances of the nonlocal operator.