Change of measure in the lookdown particle system. (English) Zbl 1302.60107
Summary: We perform various changes of measure in the lookdown particle system of P. Donnelly and T. G. Kurtz [Ann. Appl. Probab. 9, No. 4, 1091–1148 (1999; Zbl 0964.60075); Ann. Probab. 27, No. 1, 166–205 (1999; Zbl 0956.60081)]. The first example is a product type \(h\)-transform related to conditioning a generalized Fleming-Viot process without mutation on coexistence of some genetic types in remote time. We give a pathwise construction of this \(h\)-transform by just “forgetting” some reproduction events in the lookdown particle system. We also provide an intertwining relationship for the Wright-Fisher diffusion and make explicit the associated pathwise decomposition. The second example, called the linear or additive \(h\)-transform, concerns a wider class of measure-valued processes with spatial motion. Applications include a simple description of the additive \(h\)-transform of the generalized Fleming-Viot process and an immortal particle representation for the additive \(h\)-transform of the Dawson-Watanabe process.
MSC:
| 60J25 | Continuous-time Markov processes on general state spaces |
| 60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
Keywords:
Doob \(h\)-transform; Fleming-Viot process; Wright-Fisher diffusion; superprocesses; lookdown particle systemReferences:
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