The Feller Property for Generalized Branching Processes with Resurrection

We first establish a criterion for the minimal Q-function to be a Feller transition function when Q is a quasi-monotone q-matrix. We then apply this result to generalized branching q-matrices and obtain the corresponding Feller criteria for generalized branching processes. In particular, it is shown that there always exists a separating point \theta_0 with 1\leq\theta_0\leq2 or \theta_0<+\infty such that whether the generalized branching processes (with resurrection) are Feller processes or not according to \theta<\theta_0 or \theta>\theta_0, where \theta is the nonlinear number given in the branching q-matrix