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 M = sup t ≥ 0 max 1 ≤ k ≤ N ( t ) Ξ k ( t ) to be finite. Here Ξ k ( t ) is the position of the k th particle, and N ( t ) is the size of the population at time t . For some classes of processes (smooth branching diffusions with Feller-type boundary points), this results in a criterion stated in terms of the linear ODE σ 2 ( x ) 2 f ″ ( x ) + a ( x ) f ′ ( x ) = λ ( x ) ( 1 − k ( x ) ) f ( x ) . Here σ ( x ) and a ( x ) are the diffusion coefficient and the drift of the one-particle diffusion, respectively, and λ ( x ) and k ( x ) the intensity of branching and the expected number of offspring at point x , respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integral μ ( x ) ∫ π ( x , d y ) ( f ( y ) − f ( x ) ) and the product λ ( x ) ( 1 − k ( x ) ) f ( x ) , where λ ( x ) and k ( x ) are as before, μ ( x ) is the intensity of jumping at point x , and π ( x , d y ) is the distribution of the jump from x to y .