Boundedness of one-dimensional branching Markov processes

A general model of a branching Markov process on ℝ is considered. Sufficient and necessary conditions are given for the random variable M =sup max Ξ_k(t) t≥0 1≤k≤N(t) to be finite. Here Ξ_k(t) is the position of the kth 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 σ²(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 κ(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, dy) (f(y) - f(x)) and the product λ(x)(1 -κ(x))f(x) where λ(x) and κ(x) are as before, μ(x) is the intensity of jumping at point x, and π(x, dy) is the distribution of the jump from x to y.