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

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.
*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;

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 extensions (Griffin and Steel 2006; Dunson and Park 2008)

## References

*Journal of Applied Probability*48 (2): 322β32.

*The Annals of Probability*24 (2): 698β742.

*Biometrika*95 (2): 307β23.

*The Annals of Probability*21 (3): 1571β90.

*SIAM Journal on Control and Optimization*31 (2): 345β86.

*Indiana University Mathematics Journal*28 (5): 817β43.

*Journal of the American Statistical Association*101 (473): 179β94.

*Probability Theory and Related Fields*79 (2): 201β25.

*arXiv:2005.03750 [Cond-Mat, Physics:nlin, Stat]*, May.

*Mathematical Proceedings of the Cambridge Philosophical Society*54 (1): 60β71.

*Evolutionary Dynamics: Exploring the Equations of Life*. Belknap Press of Harvard University Press.

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