For the process in representation theory, see Restricted representation § Classical branching rules.
In probability theory, a branching process is a type of mathematical object known as a stochastic process, which consists of collections of random variables indexed by some set, usually natural or non-negative real numbers. The original purpose of branching processes was to serve as a mathematical model of a population in which each individual in generation produces some random number of individuals in generation , according, in the simplest case, to a fixed probability distribution that does not vary from individual to individual.[1] Branching processes are used to model reproduction; for example, the individuals might correspond to bacteria, each of which generates 0, 1, or 2 offspring with some probability in a single time unit. Branching processes can also be used to model other systems with similar dynamics, e.g., the spread of surnames in genealogy or the propagation of neutrons in a nuclear reactor.
A central question in the theory of branching processes is the probability of ultimate extinction, where no individuals exist after some finite number of generations. Using Wald's equation, it can be shown that starting with one individual in generation zero, the expected size of generation n equals μn where μ is the expected number of children of each individual. If μ < 1, then the expected number of individuals goes rapidly to zero, which implies ultimate extinction with probability 1 by Markov's inequality. Alternatively, if μ > 1, then the probability of ultimate extinction is less than 1 (but not necessarily zero; consider a process where each individual either has 0 or 100 children with equal probability. In that case, μ = 50, but probability of ultimate extinction is greater than 0.5, since that's the probability that the first individual has 0 children). If μ = 1, then ultimate extinction occurs with probability 1 unless each individual always has exactly one child.
In theoretical ecology, the parameter μ of a branching process is called the basic reproductive rate.
^Athreya, K. B. (2006). "Branching Process". Encyclopedia of Environmetrics. doi:10.1002/9780470057339.vab032. ISBN 978-0471899976.
In probability theory, a branchingprocess is a type of mathematical object known as a stochastic process, which consists of collections of random variables...
statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion process is stochastic in...
(OS) Processing (programming language), an open-source language and integrated development environment In probability theory: Branchingprocess, a Markov...
Markov processes, Lévy processes, Gaussian processes, random fields, renewal processes, and branchingprocesses. The study of stochastic processes uses...
is called the branching ratio. Thus viewing some arrivals as descendants of earlier arrivals, we have a Galton–Watson branchingprocess. The number of...
statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it is used to describe...
Branching may refer to: Branching (linguistics), the general tendency towards a given order of words within sentences and smaller grammatical units within...
process is a spatial expansion of the Galton–Watson process. Its continuous equivalent is called branching Brownian motion. An example of branching random...
discrete probability distribution, arising in contexts including branchingprocesses and queueing theory. It is named after the French mathematician Émile...
diffusion model, introduced by Paul and Tatyana Ehrenfest in 1907, and a branchingprocess, introduced by Francis Galton and Henry William Watson in 1873, preceding...
every stationary process in N outcomes is a Bernoulli scheme, and vice versa. Bessel process Birth–death processBranchingprocessBranching random walk Brownian...
sympodial growth. The pattern is similar to dichotomous branching; it is characterized by branching along stems or hyphae. In botany, sympodial growth occurs...
radix trees for random data, and trees of variable size generated by branchingprocesses. For random trees that are not necessarily binary, see random tree...
In particle physics and nuclear physics, the branching fraction (or branching ratio) for a decay is the fraction of particles which decay by an individual...
stochastic process on R × R d {\displaystyle \mathbb {R} \times \mathbb {R} ^{d}} that is usually constructed as a special limit of near-critical branching diffusions...
bronchus. Each bronchus branches into bronchioles. The branching is a result of the tip of each tube bifurcating. The branchingprocess forms the bronchi,...
polymerization, Arp2/3 complexes that give rise to actin branching and capping proteins. Due to the branchingprocess and the density of the actin cortex, the cortical...
superprocesses. Informally, superprocesses are the scaling limit of branchingprocesses, except each particle splits and dies at infinite rates. The Brownian...
random greedy algorithm which was proposed by Joel Spencer. He used a branchingprocess to formally prove the optimal achievable bound under some side conditions...
non-integrin receptor dystroglycan negatively regulates this side branchingprocess in case of cancer. These complex "Yin-yang" balancing crosstalks between...
functions of the variables. A one-dimensional GRF is also called a Gaussian process. An important special case of a GRF is the Gaussian free field. With regard...
financial software company, to boost its continual growth and the branchingprocess. "Awash Bank – Nurturing Like The River". www.awashbank.com. Retrieved...
using the most frequent tree output as the overall classification Branchingprocess, a model of a population in which each individual has a random number...
PMMA plastic are considerably different, the branching discharges turn out to be related. The branching forms taken by natural lightning also have fractal...