One way to visualize a Brownian motion process is as the limit of symmetric random walks: Let {Zn,n≥1} be the symmetric random walk on the integers. If we now speed the process up and scale the jumps accordingly we get a Brownian motion process in the limit. More precisely, suppose we jump every Dt and make a jump of size Dx. if we let Z(t) denote the position of the process at time t then
The Brownian motion process plays a role in the theory of stochastic processes similar to the role of the normal distribution in the theory of random variables.
The name Brownian motion comes from a phenomenon first observed by the British botanist Robert Brown in 1828. He looked at pollen grains suspended in water under the microscope and observed that they perform a continual swarming motion. This phenomenon was first explained by Einstein in 1905 who said the motion comes from the pollon being hit by the molecules in the surrounding water. The mathematical derivation of the Brownian motion process was first done by Wiener in 1918, and in his honor it is often called the Wiener process.
If s=1 the process is called standard Brownian motion.
In brown with which==1, we draw sample paths of a standard Brownian motion process.
Here are some properties of Brownian motion:
1) BM will eventually hit any and every real value, no matter how large or how negative! It may be a million units above the axis, but it will (wp1) be back down again to 0, by some later time.
2) Once BM hits zero (or any particular value), it immediately hits it again infintely often, and then again from time to time in the future.
3) Spatial Homogeniety: B(t) + x for any x 
R is a BM started at x.
4) Symmetry: -B(t) is a Brownian motion
5) Scaling: √c•B(t/c) for any c > 0 is a BM
6) Time inversion:
is a BM.
7) BM is time reversible
8) BM is self-similar (that is its paths are fractals):
Consider the four graphs of BM paths drawn in brown with which==2. They are drawn without labeling on the axis. They appear completely the same, but if we add the labels (which==3) we see that the scales are completely different. This phenomena is called self-similarity.
Brownian Motion is an Example of a process that has a fractal dimension of 2. One of its occurrences is in microscopic particles and is the result of random jostling by water molecules (if water is the medium). So in moving from a given location in space to any other, the path taken by the particle is almost certain to fill the whole space before it reaches the exact point that is the 'destination' (hence the fractal dimension of 2).
9) The last property of BM is so much fun we will give it its own section:
S. Then f(t) = X(t,w) is a function. (Usually we supress w, though).
In the case of BM, what are the properties of a typical realization B(t)? First let's look at continuity:
Now by the definition we have that B(t+h)-B(t) ~ N(0,h), therefore E[(B(t+h)-B(t))2] = h and so the size of an increment of |B(t+h)-B(t)| is about √h. So as h→0 √h→0 which implies continuity.
How about differentiability? Now we have
and we see that BM is nowhere differentiable!
(Of course this is rather heuristic but it can be made rigorous).
The idea of functions that are contiuous but nowhere differentiable has a very interesting history. It was first discussed in 1806 by André Marie Ampère and trying to show that such a function exists was one of the main open problems during the 19th century. More than fifty years later it was Karl Theodor Wilhelm Weierstrass who finally succeeded in constructing such a function. as follows:
weierstrass draws the function for b = .2 and a = 5 + 7.5p(and a finite sum!)
The hard part here was not the construction but to show that the function existed! For the proof he developed was is now known as the Stone-Weierstrass theorem.
Shorty after that a new branch of mathematics called functional analysis was developed. It studies the properties of real-valued functions on function space. Here are some Example s of such functionals:
Of course one needs to specify the space of functions fow which a certain functional applies. Standard "function spaces" are C, the space of all continuous functions and C1, the space of all continuous functions with a continuous derivative.
One of the results of functional analysis is that C is much larger than C1, actually of a higher order of infinity. So consider the following "experiment": pick any continuous function in C. Then the probability that it has a continuous derivative anywhere is 0! So functions such as Weierstrass (or the paths of BM) are not the exception, they are the rule. Or, all the functions we study in mathematics are completely irrelevant in nature.