Brownian Motion and Diffusion Equations

History of Brownian Motion

Discovered by Robert Brown, 1827 Found that small particles suspended

in liquid moved about randomly Guoy discovered that particle motion

was caused collisions of molecules In 1905, Einstein developed a

mathematical model for Brownian Motion

Discrete Model Brownian Motion Consider N discrete, independent steps in

which a particle will move right with probability p, or to the left with probability 1-p.

Clearly, the number of right steps the particle takes is binomially distributed with parameters p and N. The number of left steps is binomially distributed with parameters 1-p and N.

The final position of the particle, the number of right steps minus left steps, has an expectation N(1-2p)

Binomial curve approximates to normal curve for large values of N and many trials, yielding…

Histogram of 1000 p=.5 random walks with 15 steps

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Abstract Construction of a Brownian Motion

A function X(t) is a Brownian Motion iff:

1) The mechanism producing random variations does not change with time. (ie, identical motions)

2) All time intervals are mutually independent

3) X(0) =0 and X(t) is a continuous function of t

Questions???