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A stochastic process is one whose behavior is non-deterministic in that the next state of the environment is not fully determined by the previous state of the environment.
In mathematics, specifically in probability theory, the field of stochastic processes has for some decades been a major area of research, to which hundreds of researchers have devoted their careers. See that article for more.
A stochastic matrix is a matrix that has non-negative real entries that sum to 1 in each column.
In artificial intelligence stochastic programs work by using probabilistic methods to generate emergent properties, that is, characteristics that were not explicitly programmed in. A deterministic environment is much simpler for an agent to deal with.
An example of a stochastic process in the natural world is pressure in a gas. Even though each molecule is moving deterministically, a collection of them is unpredictable (this is an example of chaos out of order). A large enough set of molecules will exhibit stochastic characteristics, such as filling the container, exerting equal pressure, diffusing along concentration gradients, etc. These are emergent properties of the system.
In music stochastic elements are randomly generated elements created by strict mathematical processes.
Stochastic processes can be used in music either to compose a fixed piece, or produced in performance. Stochastic music was pioneered by Iannis Xenakis, who used probability, game theory, group theoryAbstract algebra Group theory Group theory is that branch of mathematics concerned with the study of groups. Please refer to the Glossary of group theory for the definitions of terms used throughout group theory. See also list of group theory topics., set theorySet theory is the mathematical theory of sets, which represent collections of abstract objects. It has a central role in modern mathematical theory, providing the basic language in which most of mathematics is expressed. For more information on set theory, and Boolean algebraIn mathematics and computer science, Boolean algebras or Boolean lattices are algebraic structures which "capture the essence" of the logical operations AND, OR and NOT as well as the corresponding set theoretic operations intersection, union and compleme, and frequently used computertower of a personal computer. A computer is a device for making calculations or controlling operations that are expressible in numerical or logical terms. While factually accurate, this definition and those found in other dictionaries are so broad that ths to produce his scores. Earlier, John CageJohn Milton Cage ( September 5, 1912 August 12, 1992) was an experimental music composer and writer, possibly best known (some might say notorious) for his piece 4 33 often described (somewhat erroneously) as "four and a half minutes of silence. He was an and others had composed aleatoric musicAleatoric (or aleatory music or composition is music where some element of the composition is left to chance. The term was devised by the French composer Pierre Boulez to describe works where the performer was given certain liberties with regard to the or, which is created by chance processes but does not have the strict mathematical basis (Cage's Music of Changes, for example, uses a system of charts based on the I-Ching).