Review for Week of Apr 17

We learned that the gap between two points of the random walk:

• Has a uniform metric
• Has an upper limit with probability 1
• Has a lower limit with probability 1
• converges to a limit
• Limit is continuous
• There exists a limiting function (which is continuous) that is the sum of the telescoping zooms
• zoom gets wilder and wilder as delta-t gets smaller
• Continuous parts of limiting function are cauchy
• Limiting function converges
• => Brownian Motion exists
• And the paths are fractals
• Length of path is infinite
• Time it takes to travel path is unit 1

This is so weird. You go from point A to point B, you never diverge from a direct path by more than a certain limit, and yet the distance you travel is infinite.

I thought for a while that maybe we were looking at points that represented two aspects of travel like x is distance and y is time as opposed to graphing that actual location in 2D that our pollen particle theoretically is moving in. I think we are looking at the actual path, which only makes this more interesting.