On the construction and support properties of measure-valued diffusions on D ⊆ Rd with spatially dependent branching

• 成果类型: Article
• 署名作者: Engländer, J; Pinsky, RG
• 署名单位: Technion Israel Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1022677383
• 发表日期: 1999
• 页码: 684-730
• 关键词: weighted occupation time; EQUATIONS; uniqueness

摘要

In this paper, we construct a measure-valued diffusion on D subset of or equal to R-d whose underlying motion is a diffusion process with absorption at the boundary corresponding to an elliptic operator L = 1/2 del . a del + b . del on D subset of or equal to R-d and whose spatially dependent branching term is of the form beta(x)z - alpha(x)z(2), x is an element of D, where beta satisfies a very general condition and alpha > 0. In the case that alpha and beta are bounded from above, we show that the measure-valued process can also be obtained as a limit of approximating branching particle systems. We give criteria for extinction/survival, recurrence/transience of the support, compactness of the support, compactness of the range, and local extinction for thc? measure-valued diffusion. We also present a number of examples which reveal that the behavior of the measure-valued diffusion may be dramatically different from, that of the approximating particle systems.