Brownian Motion

For the past couple weeks in class we have spent a lot of time discussing (and trying to prove) Brownian motion and it's characteristics. While very interesting, this topic came off to me as quite abstract and difficult to picture in my head. It inolves a lot of probability and took some statistical background knowledge to follow the lecture.

The first subtopic which we came across that I had some trouble understanding was "random walk". After I researched this a little bit I found that it is also called a "drunkards walk" and looks something like this:

1. Start at some given point
2. Move forward to another point in some direction (chosen at random) with each direction having the same probability.

Here is an example showing eight different "drunkards walks" with 100 timesteps (points) each:

Not as symmetric or pleasing to the eye as the other graphics we've looked at prior too this section, but important nonetheless.

I also took some time to try and learn to make a random walk using Mathematica, and will post my results (*here*) when I finish.

From what I can tell, Brownian motion is simply a specific type of random walk, which has to do with the diffusion process of certain particles. I found this graphic of Brownian motion:

Once again not the most pleasing to the eye, but I guess that's why its "random" motion. As I review I'll post the proof we composed of the existence of Brownian motion.