A general model of a branching Markov process on ℝ is considered.
Sufficient and necessary conditions are given for the random variable
M=supt≥0max1≤k≤N(t)Ξk(t)
to be finite. Here Ξk(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 ODEσ2(x)2f″(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,dy)(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,dy) is the distribution of the jump from x to y.