Notebooks
http://bactra.org/notebooks
Cosma's NotebooksenBranching Processes
http://bactra.org/notebooks/2023/07/27#branching-processes
<P>A class of <a href="stochastic-processes.html">stochastic process</a>
important as models in genetics and population biology, chemical kinetics, and
filtering. The basic idea is that there are a number of objects, often called
particles, which, in some random fashion, reproduce ("branch") and die out;
they can be of multiple types and occupy
differing <a href="spatial-statistics.html">spatial locations</a>. They can
pursue their trajectories and their biographies either independently, or with
some kind of statistical dependence across particles.
<P>The most basic version has one type of particle, and no spatial
considerations. At each time step, each parrticle gives rise to a random
number of offspring; the distribution of offspring is fixed, and the number is
independent across time-steps and across lineages (IID). This is the so-called
Galton-Watson branching process. Galton introduced it as a model of the
survival of (patrilneal) family names, so that only male offspring counted; he
required the distribution of time until a given lineage went extinct. This was
provided almost immediately by Watson, in a very elegant use of the method of
generating functions, which is, itself, reproduced in probability textbooks
down to the present day. (However, when I first encoutnered the problem, in a
probability class, the teacher presented it as one about the survival
of <em>matrilineal</em> lineages, defined by inheritance of mitochondrial DNA.
Whether this was conscious subversion of the patriarchy, or just a reflection
of the changing scientific interests between the 1890s and the 1990s, I
couldn't say.)
<P>See also:
<a href="compartment-models.html">Compartment Models</a>;
<a href="epidemic-models.html">Epidemic Models</a>;
<a href="social-contagion.html">Social Contagion</a>
<ul>Recommended (introductory):
<li>Geoffrey Grimmett and David Stirzaker, <cite>Probability and Random
Processes</cite> [This is my favorite probability textbook, and returns to
branching processes in many places.]
</ul>
<ul>Recommended (forbiddingly technical):
<li>P. Del Moral and L. Miclo, "Branching and Interacting Particle
Systems Approximations of Feynman-Kac Formulae with Applications to Nonlinear
Filtering", in J. Azema, M. Emery, M. Ledoux and M. Yor
(eds)., <cite>Semainaire de Probabilites XXXIV</cite> (Springer-Verlag, 2000),
pp. 1--145 [<a href="http://math1.unice.fr/~delmoral/seminaire.ps">Postscript
preprint</a>. Looks like a trial run for Del Moral's book, below, which I've
yet to read.]
</ul>
<P>To read:
<li>David Assaf, Larry Goldstein and Ester Samuel-Cahn, "An unexpected
connection between branching processes and optimal stopping", <cite>Journal of
Applied Probability</cite> <strong>37</strong> (2000):
613--6, <a href="http://arxiv.org/abs/math.PR/0510587">math.PR/0510587</a>
[This sounds like a nice pedagogical topic for a course in stochastic processes. I teach a course in stochastic processes....]
<li>Michael Assaf and Baruch Meerson, "Spectral Theory of Metastability
and Extinction in Birth-Death
Systems", <a href="http://dx.doi.org/10.1103/PhysRevLett.97.200602">Physical
Review Letters</cite> <strong>97</strong> (2006): 200602</a>,
<a href="http://arxiv.org/abs/cond-mat/0610415">cond-mat/0610415</a>
<li>Krishna B. Athreya, <cite>Branching Processes</cite>
<li>K. B. Athreya, A.P. Ghosh, S. Sethuraman, "Growth of preferential
attachment random graphs via continuous-time branching
processes", <a href="http://arxiv.org/abs/math.PR/0701649">math.PR/0701649</a>
<li>Ellen Baake, Hans-Otto Georgii, "Mutation, selection, and ancestry
in branching models: a variational
approach", <a href="http://arxiv.org/abs/q-bio.PE/0611018">q-bio.PE/0611018</a>
<li>Romulus Breban, Raffaele Vardavas and Sally Blower,
"Linking population-level models with growing networks: A class of epidemic models", <a href="http://dx.doi.org/10.1103/PhysRevE.72.046110"><cite>Physical Review E</cite> <strong>72</strong> (2005): 046110</a>
<li>Nicolas Champagnat, Régis Ferrière, Sylvie
Méléar, "Individual-based probabilistic models of adaptive
evolution and various scaling
approximations", <a href="http://arxiv.org/abs/math.PR/0510453">math.PR/0510453</a>
<li>Charles R. Doering, Khachik V. Sargsyan and Leonard M. Sander,
"Extinction times for birth-death processes: exact results, continuum
asymptotics, and the failure of the Fokker-Planck approximation", <a
href="http://arxiv.org/abs/q-bio/0401016">q-bio/0401016</a>
<li>Pierre Del Moral, <cite>Feynman-Kac Formulae: Genealogical and
Interacting Particle Systems</cite> [This looks <em>really, really cool</em>]
<li>Janos Englander, "Branching diffusions, superdiffusions and random media", <cite>Probability Surveys</cite> <strong>4</strong> (2007): 303--364,
<a href="http://arxiv.org/abs/0710.0236">arxiv:0710.0236</a>
<li>Vicenc Gomez, Hilbert J. Kappen and Andreas Kaltenbrunner,
"Modeling the structure and evolution of discussion cascades", <a href="http://arxiv.org/abs/1011.0673">arxiv:1011.0673</a>
<li>P. Haccou et al., <cite>Branching Processes: Variation, Growth,
and Extinction of Populations</cite>
<li>Jose Luis Iribarren and Esteban Moro, "Branching Dynamics of Viral
Information Spreading", <<a href="http://dx.doi.org/10.1103/PhysRevE.84.046116">cite>Physical Review E</cite> <strong>84</strong> (2011): 046116</a>
<li>Predrag R. Jelenkovic, Jian Tan, "Modulated Branching Processes,
Origins of Power Laws and Queueing
Duality", <a href="http://arxiv.org/abs/0709.4297">0709.4297</a>
<li>Junghyo Jo, Jean-Yves Fortin, M. Y. Choi, "Weibull-type limiting distribution for replicative systems", <cite>Physical Review E</cite> <strong>83</strong> (2011): 031123, <a href="http://arxiv.org/abs/1103.3038">arxiv:1103.3038</a>
<li>Jean-Francois Le Gall, <cite>Spatial Branching Processes,
Random Snakes and Partial Differential Equations</cite>
<li>Brendan P. M. McCabe1, Gael M. Martin, David Harris, "Efficient probabilistic forecasts for counts", <a href="http://dx.doi.org/10.1111/j.1467-9868.2010.00762.x"><cite>Journal
of the Royal Statistical Society</cite> B <strong>73</strong> (2011): 253--272</a>
<li>Sebastian Müller, "Strong recurrence for branching Markov
chains", <a href="http://arxiv.org/abs/0710.4651">arxiv:0710.4651</a>
<li>Fabricio Murai, Bruno Ribeiro, Don Towsley, Krista Gile, "Characterizing Branching Processes from Sampled Data", <a href="http://arxiv.org/abs/1302.5847">arxiv:1302.5847</a>
<li>Victor M. Panaretos, "Partially observed branching processes for
stochastic
epidemics", <a href="http://dx.doi.org/10.1007/s00285-006-0062-6"><cite>Journal
of Mathematical Biology</cite> <strong>54</strong> (2007): 645--668</a>
<li>Su-Chan Park, Joachim Krug, Léo Touzo & Peter Mörters, "Branching with Selection and Mutation I: Mutant Fitness of Fréchet Type",
<a href="https://doi.org/10.1007/s10955-023-03125-3"><cite>Journal of Statistical Physics</cite> <strong>190</strong> (2023): 115</a>
<li>David Sankoff, "Branching Processes with Terminal Types:
Application to Context-Free Grammars", <a
href="http://www.jstor.org/pss/3211893"><cite>Journal of Applied
Probability</cite> <strong>8</strong> (1971): 233--240</a>
<li>D. Sornette and S. Utkin, "Limits of declustering methods for disentangling exogenous from endogenous events in time series with foreshocks, main shocks, and aftershocks", <cite>Physical Review E</cite> <strong>79</strong> (2009): 061110, <a href="http://arxiv.org/abs/0903.3217">arxiv:0903.3217</a>
</ul>