Boundedness of one-dimensional branching Markov processes. (1997)

• Record Type: Journal Article
• Authors: Karpelevich, F. I.; Suhov, Yu. M.
• Is Part Of: Journal of applied mathematics and stochastic analysis. Volume 10:Number 4(1997)
• Page Start: 307
• Page End: 332
• Publication Date: 1997
• Subjects: Mathematical models -- Periodicals; Computer simulation -- Periodicals; Computer science -- Mathematics -- Periodicals; Computer science -- Mathematics; Computer simulation; Mathematical models; Applied Mathematics; Periodicals; Electronic journals; 519.22
• DOI: 10.1155/S1048953397000397
• Languages: English
• ISSNs: 1048-9533

Abstract

Abstract : A general model of a branching Markov process on ℝ 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 .