\(\renewcommand{\var}{\operatorname{Var}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\pd}{\partial} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\bf}[1]{\mathbf{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\mm}[1]{\mathrm{#1}} \renewcommand{\cc}[1]{\mathcal{#1}} \renewcommand{\oo}[1]{\operatorname{#1}} \renewcommand{\gvn}{\mid} \renewcommand{\II}{\mathbb{I}}\)

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 – 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)

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.

Dunson, David B, and Ju-Hyun Park. 2008. “Kernel Stick-Breaking Processes.” *Biometrika* 95 (2): 307–23. https://doi.org/10.1093/biomet/asn012.

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.

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

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.