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Discrete-measure valued stochastic processes

Usefulness: 🔧
Novelty: 💡
Uncertainty: 🤪 🤪 🤪
Incompleteness: 🚧 🚧 🚧

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Stochastic processes indexed by time whose state is a discrete (possibly only countable) measure. Popular in, for example, mathematical models of alleles in biological evolution.

Population genetics keywords, in approximate order of generality and chronology: Fisher-Wright diffusion, Moran process, Viot-Fleming process. Obviously there are many other processes matching this broad description.

For one example, any element-wise-positive vector-valued stochastic process can be made into a discrete-measure-valued process by normalising the state vector to sum to 1. For another, count time series are realisations of the measures of such as process of these. Chinese-restaurant processes as processes in time presumably fit here, although AFAICT the use of these processes in the literature is usually not the time-evolving construction, but rather the infinite-time limit of such a process, which is confusing nomenclature.

If the process does not take values in discrete measures, that would be a different notebook, which does not at the moment exist; For now, I note that state filters induce such processes, although in an inference setting rather than a purely probabilistic one. The interacting particle systems are important in that context, too. Also, for example, stochastic differential equations are also measure-valued stochastic processes – once again, usually not over discrete measures.

The Dirichlet process, despite its name is not usually sampled as a time process and in any case has boring dynamics. However, it does I believe have time-indexed exnsions (???; ???)

Refs

Asselah, Amine, Pablo A. Ferrari, and Pablo Groisman. 2011. “Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces.” Journal of Applied Probability 48 (2): 322–32. https://doi.org/10.1239/jap/1308662630.

Donnelly, Peter, and Thomas G. Kurtz. 1996. “A Countable Representation of the Fleming-Viot Measure-Valued Diffusion.” The Annals of Probability 24 (2): 698–742. https://www.jstor.org/stable/2244946.

Ethier, S. N., and R. C. Griffiths. 1993. “The Transition Function of a Fleming-Viot Process.” The Annals of Probability 21 (3): 1571–90. https://www.jstor.org/stable/2244588.

Ethier, S. N., and Thomas G. Kurtz. 1993. “Fleming–Viot Processes in Population Genetics.” SIAM Journal on Control and Optimization 31 (2): 345–86. https://doi.org/10.1137/0331019.

Fleming, Wendell H, and Michel Viot. 1979. “Some Measure-Valued Markov Processes in Population Genetics Theory.” Indiana University Mathematics Journal 28 (5): 817–43. https://www.jstor.org/stable/24892583.

Konno, N., and T. Shiga. 1988. “Stochastic Partial Differential Equations for Some Measure-Valued Diffusions.” Probability Theory and Related Fields 79 (2): 201–25. https://doi.org/10.1007/BF00320919.

Moran, P. a. P. 1958. “Random Processes in Genetics.” Mathematical Proceedings of the Cambridge Philosophical Society 54 (1): 60–71. https://doi.org/10.1017/S0305004100033193.

Nowak, M. A. 2006. Evolutionary Dynamics: Exploring the Equations of Life. Cambridge, Mass: Belknap Press of Harvard University Press.