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Brownian motion

There are two meanings of the term Brownian motion: the physical phenomenon
that minute particles immersed in a fluid will experience a random movement,
and one of the mathematical models used to describe it.

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, and another
important example is the evolution of physical characteristics in the fossil
record.

Brownian motion is the simplest stochastic process 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 actual accuracy as models that dictates
their use. All three quoted examples of Brownian motion are cases of this:
it has been argued that Levy flights are a more accurate, if still
imperfect, model of stock-market fluctuations; the physical Brownian motion
can be modelled more accurately by more general diffusion process; and the
dust hasn't settled yet on what the best model for the fossil record is,
even after correcting for non-gaussian data.

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, and he observed them 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 none other than Albert Einstein in 1905.

At that time the atomic nature of matter was still a controversial idea.
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.

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 σ2 dt; the parameter μ is the drift velocity,
and the parameter σ2 is the power of the noise. 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 time-homogeneous stochastic process
with independent increments and which is continuous in probability. 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 Brownian motion with drift.

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
Fokker-Planck equation depending on whether it is formulated in terms of
random trajectories or probability densities.
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