Two and one dimensional Brownian motion


Brownian motion originally refers to the random motion observed under microscope of a pollen immersed in water. Einstein pointed out that this motion is caused by random bombardment of (heat excited) water molecules on the pollen. Modern theory call it a stochastic process. An approximation (discretization) of the 2 dimensional Brownian motion can be described as a "drunken man wandering around the square". More precisely, each of his steps (in both x- and y- directions) are independent normal random variables. In fact this is what the simulation is performing.

How to use this applet:

The applet shows a square on the left, the 2 dimensional Brownian Motion starts at the center. On the right, a 1 dimensional Brownian Motion is plotted (just the y position on the 2 dim picture plot against time). Click on "Start" to begin. Click "start" again to a new piece of paper. After a while, the Brownian motion will probably be outside of the square (in fact, almost surely). The theory also say that it will return to the center with probability one, but do not hold your breath for it---the expected time before return to the origin is infinity. Another interesting fact is that the path he follows is continuous but no where differentiable.

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Written by Mai Zhou Copyright 1998 Mai Zhou. All rights reserved.         back to my applet index page