# Discrete-measure valued stochastic processes

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Uncertainty: đź¤Ş đź¤Ş đź¤Ş
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$$\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}$$

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 â€“ 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 (Griffin and Steel 2006; Dunson and Park 2008)

# Refs

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

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

Dunson, David B, and Ju-Hyun Park. 2008. â€śKernel Stick-Breaking Processes.â€ť Biometrika 95 (2): 307â€“23. https://doi.org/10.1093/biomet/asn012.

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

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

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

Griffin, J. E., and M. F. J. Steel. 2006. â€śOrder-Based Dependent Dirichlet Processes.â€ť Journal of the American Statistical Association 101 (473): 179â€“94. https://doi.org/10.1198/016214505000000727.

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

Moran, P. a. P. 1958. â€śRandom Processes in Genetics.â€ť Mathematical Proceedings of the Cambridge Philosophical Society 54 (1): 60â€“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.