A jiggling motion that can be observed, under a microscope, in tiny particles in water and in smoke particles in air. This jiggling is the result of the movements of water and air molecules. At room temperature, the unpredictable movements of water molecules are strong enough to bounce much larger particles around, which is visible under a microscope, if not to the naked eye. Discovered in 1827 by Scottish botanist Robert Brown.

Einstein gave an axiomatic definition of Brownian motion, in use in mathematical physics, statistical mechanics and probability theory:
(1-dimensional) Brownian motion is the probability measure on the set of continuous functions B: R -> R such that
1. For all nonintersecting intervals (a,b) and (c,d), the random variables B(b)-B(a) and B(d-c) are independent.
2. For all intervals (a,b), the variable B(b)-B(a) has a normal distribution with variance b-a.
This captures our intuitions of what a "random" motion of a particle should be. Unfortunately, he forgot to prove that such a distribution exists!

Actual constructions (obviously only in the sense of mathematical analysis, not in the sense of mathematical logic, if you're into that sort of thing) only came later; Paul Lévy and Wiener provided correct constructions.

In fact, even slight changes to the definition can yield nonexistent results! For instance, Durrett provides an example where replacing the normal distribution by a Cauchy distribution can be proven not to exist: attempting to follow through the construction yields a function that is almost always nowhere continuous.

To get multidimensional Brownian motion, it is sufficient to take independent Brownian motions along each axis; the unique property of the multidimensional normal distribution show that this is a good idea, and the result independent of the particular choice of axes.

Brownian motion is a very important example of a continuous stochastic process. Obviously, it's a continuous martingale.

Stochastic integrals are an attempt to define an integral with respect to the "measure" dB(s), in the sense of Stieltjes integrals.

I fully understand Brownian Motion following the three days I have been on this everything2 site.  Randomness is rarely random for sure.  The mere and shear force of the concepts bouncing off of the inside of my concious and unconcious skull has kept me awake and forced me to respond and I do not even know how to respond properly.  I will be the talk of the regular "clear out the junk meeting" at E2.

The pain in my head, more properly, the pain in my mind is not lessening.  It is becoming larger, not worse or stronger.  A smashed fingernail hurts until it explodes, then the feeling is different until it does not stop bleeding and you get lightheaded.  Dad would say, "Soak it in Mennen aftershave and it will not hurt tomorrow".  The comma you fall into will last way past tomorrow.