Boundedness of one-dimensional branching Markov processes

[摘要] A general model of a branching Markov process onℝis considered. Sufficient and necessary conditions are given for the random variableM=supt≥0max1≤k≤N(t)Ξk(t)to be finite. HereΞk(t)is the position of thekth particle, andN(t)is the size of the population at timet. For some classes of processes (smooth branching diffusions with Feller-type boundary points), this results in acriterion stated in terms of the linearODEσ2(x)2f″(x)+a(x)f′(x)=λ(x)(1−k(x))f(x). Hereσ(x)anda(x)are the diffusion coefficient and the drift of the one-particle diffusion, respectively, andλ(x)andk(x)the intensity of branching and the expected number of offspring at pointx, respectively. Similarly, for branching jump Markov processes the conditions are expressed in terms of the relations between the integralμ(x)∫π(x,dy)(f(y)−f(x))and the productλ(x)(1−k(x))f(x), whereλ(x)andk(x)are as before,μ(x)is the intensity of jumping at pointx, andπ(x,dy)is the distribution of the jump fromxtoy.

[学科分类] 应用数学