The Annals of Probability

Wandering Random Measures in the Fleming-Viot Model

Donald A. Dawson and Kenneth J. Hochberg

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Abstract

Fleming and Viot have established the existence of a continuous-state-space version of the Ohta-Kimura ladder or stepwise-mutation model of population genetics for describing allelic frequencies within a selectively neutral population undergoing mutation and random genetic drift. Their model is given by a probability-measure-valued Markov diffusion process. In this paper, we investigate the qualitative behavior of such measure-valued processes. It is demonstrated that the random measure is supported on a bounded generalized Cantor set and that this set performs a "wandering" but "coherent" motion that, if appropriately rescaled, approaches a Brownian motion. The method used involves the construction of an interacting infinite particle system determined by the moment measures of the process and an analysis of the function-valued process that is "dual" to the measure-valued process of Fleming and Viot.

Article information

Source
Ann. Probab., Volume 10, Number 3 (1982), 554-580.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993767

Digital Object Identifier
doi:10.1214/aop/1176993767

Mathematical Reviews number (MathSciNet)
MR659528

Zentralblatt MATH identifier
0492.60045

JSTOR
links.jstor.org

Subjects
Primary: 60G57: Random measures
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 60J25: Continuous-time Markov processes on general state spaces 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 92A15

Keywords
Measure-valued Markov process random measure Hausdorff dimension ladder or stepwise-mutation model population genetics Fleming-Viot model wandering coherent distribution

Citation

Dawson, Donald A.; Hochberg, Kenneth J. Wandering Random Measures in the Fleming-Viot Model. Ann. Probab. 10 (1982), no. 3, 554--580. doi:10.1214/aop/1176993767. https://projecteuclid.org/euclid.aop/1176993767


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