Boundedness of one-dimensional branching Markov processes
A general model of a branching Markov process on is considered. Sufficient and necessary conditions are given for the random variable to be finite. Here is the position of the th particle, and is the size of the population at time . For some classes of processes (smooth branching diffusions with Feller-type boundary points), this results in a criterion stated in terms of the linear . Here and are the diffusion coefficient and the drift of the one-particle diffusion, respectively, and and the intensity of branching and the expected number of offspring at point , respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral and the product , where and are as before, is the intensity of jumping at point , and is the distribution of the jump from to .