<P>The term <I><B>Brownian motion</B></I> (in honor of the botanist <a href="http://en.wikipedia.org/wiki/Robert_Brown_%28botanist%29" target="_blank" >Robert Brown</A>) refers to either</P>
<OL>
<LI>The physical phenomenon that minute particles, immersed in a fluid, move about randomly; or
<LI>The mathematical models used to describe those random movements. </LI></OL>
<P>The mathematical model can also be used to describe many phenomena not resembling (other than mathematically) the random movements of minute particles. An often quoted example is <a href="http://en.wikipedia.org/wiki/Stock_market" target="_blank" >stock market</A> fluctuations. Another example is the evolution of physical characteristics in the fossil record.</P>
<P>Brownian motion is among the simplest <a href="http://en.wikipedia.org/wiki/Stochastic_process" target="_blank" >stochastic processes</A> on a continuous domain, and it is a <a href="http://en.wikipedia.org/wiki/Limit_%28mathematics%29" target="_blank" >limit</A> of both simpler (see <a href="http://en.wikipedia.org/wiki/Random_walk" target="_blank" >random walk</A>) and more complicated stochastic processes. This <a href="http://en.wikipedia.org/wiki/Universality_%28dynamical_systems%29" target="_blank" >universality</A> is closely related to the universality of the <a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_blank" >normal distribution</A>. 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:</P>
<OL>
<LI>It has been argued that <a href="http://en.wikipedia.org/wiki/L%C3%A9vy_flight" target="_blank" >Lévy flights</A> are a more accurate, if still imperfect, model of stock-market fluctuations.
<LI>The physical Brownian motion can be modelled more accurately by a more general <a href="http://en.wikipedia.org/wiki/Diffusion" target="_blank" >diffusion process</A>.
<LI>The dust hasn't settled yet on what the best model for the fossil record is, even after correcting for non-<a href="http://en.wikipedia.org/wiki/Normal_distribution" target="_blank" >Gaussian</A> data. </LI></OL>
<P>reference: <a href="http://en.wikipedia.org/wiki/Brownian_motion" target="_blank" >http://en.wikipedia.org/wiki/Brownian_motion</A></P>
<P>random walk is just one of phenomenons mentioned in Brownian Motion, i guess. i sort of think that random walk was derived from Brownina Motion.</P>
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