Brownian motion
From Academic Kids

The term Brownian motion (in honor of the botanist Robert Brown) refers to either
 The physical phenomenon that minute particles immersed in a fluid move around randomly; or
 The mathematical models used to describe those random movements.
The mathematical model can also be used to describe many phenomena not resembling (other than mathematically) the random movement of minute particles. An often quoted example is stock market fluctuations. Another example is the evolution of physical characteristics in the fossil record.
Brownian motion is among the simplest stochastic processes on a continuous domain, and it is a limit of both simpler (see random walk) and more complicated stochastic processes. This universality is closely related to the universality of the normal distribution. In both cases, it is often mathematical convenience rather than accuracy as models that motivates their use. All three quoted examples of Brownian motion are cases of this:
 It has been argued that Lévy flights are a more accurate, if still imperfect, model of stockmarket fluctuations.
 The physical Brownian motion can be modelled more accurately by a more general diffusion process.
 The dust hasn't settled yet on what the best model for the fossil record is, even after correcting for nonGaussian data.
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History of Brownian motion
Brownian motion was discovered by the biologist Robert Brown in 1827. The story goes that Brown was studying pollen particles floating in water under the microscope. He then observed minute particles within vacuoles in the pollen grains executing the jittery motion that now bears his name. By doing the same with particles of dust, he was able to rule out that the motion was due to pollen being "alive", but it remained to explain the origin of the motion. The first to give a theory of Brownian motion was Louis Bachelier in 1900 in his PhD thesis "The theory of speculation".
At that time the atomic nature of matter was still a controversial idea. Albert Einstein observed that, if the kinetic theory of fluids was right, then the molecules of water would move at random and so a small particle would receive a random number of impacts of random strength and from random directions in any short period of time. This random bombardment by the molecules of the fluid would cause a sufficiently small particle to move in exactly the way described by Brown. Jean Perrin carried out experiments to test the new mathematical models, and his published results finally put an end to the centurylong dispute about the reality of atoms and molecules.
Description of the mathematical model
Mathematically, Brownian motion is a Wiener process in which the conditional probability distribution of the particle's position at time t+dt, given that its position at time t is p, is a normal distribution with a mean of p+μ dt and a variance of σ^{2} dt; the parameter μ is the drift velocity, and the parameter σ^{2} is the power of the noise. These properties clearly establish that Brownian motion is Markovian (i.e. it satisfies the Markov property). Brownian motion is related to the random walk problem and it is generic in the sense that many different stochastic processes reduce to Brownian motion in suitable limits.
In fact, the Wiener process is the only timehomogeneous stochastic process with independent increments that has continuous trajectories. These are all reasonable approximations to the physical properties of Brownian motion.
The mathematical theory of Brownian motion has been applied in contexts ranging far beyond the movement of particles in fluids. For example, in the modern theory of option pricing, asset classes are sometimes modeled as if they move according to a closely related process, geometric Brownian motion.
It turns out that the Wiener process is not a physically realistic model of the motion of Brownian particles. More sophisticated formulations of the problem have led to the mathematical theory of diffusion processes. The accompanying equation of motion is called the Langevin equation or the FokkerPlanck equation depending on whether it is formulated in terms of random trajectories or probability densities.
See also
 brown noise (Martin Gardner proposed this name for sound generated with random intervals. It is a pun on Brownian motion and white noise.)
 osmosis
 Brownian tree
 ultramicroscope
 Brownian ratchet
 Brownian frontier
 Brownian bridge
References
 Brown, Robert, "A brief account of microscopical observations made in the months of June, July and August, 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies." Phil. Mag. 4, 161173, 1828. (PDF version of original paper including a subsequent defense by Brown of his original observations, Additional remarks on active molecules.) (http://sciweb.nybg.org/science2/pdfs/dws/Brownian.pdf)
 Einstein, A. "Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen." Ann. Phys. 17, 549, 1905. [1] (http://www.wileyvch.de/berlin/journals/adp/549_560.pdf)
 Einstein, A. Investigations on the Theory of Brownian Movement. New York: Dover, 1956. ISBN 0486603040
 Nelson, Edward, Dynamical Theories of Brownian Motion (1967) (PDF version of this outofprint book, from the author's webpage.) (http://www.math.princeton.edu/~nelson/books.html)ca:Moviment brownià
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