Sigma-fields, Borel algebra and probability measures

The Wiener Process and Brownian Motion

Zoom in on a Wiener process, with w(0)=0 located in the middle of the picture. A demonstration of Brownian scaling, showing $V_{t}=(1/{\sqrt {c}})W_{ct}$ for decreasing c. Note that the average features of the function do not change while zooming in, and note that it zooms in quadratically faster horizontally than vertically, illustrating the fact that $dw \sim \sqrt{dt}$ $dw \sim \sqrt{dt}$ . From Wikipedia.org

A video about Brownian motion
Ilustration of Brownian Motion, from Wikipedia:
Brownian motion on Wikipedia
The fluctuation-dissipation theorem on wikipedia
Einstein's 1905 paper about Brownian motion
One minute summary of the paper
In the lecture we mention mathematical properties a model of Brownian motion should satisfy. There are different ways to prove that such a stochastic process exists. One way is to use the (nontrivial) Kolmogorov extension theorem.
Mathematical details: A short but rigorous introduction to Brownian Motion, by Andy Dahl.
Infinite divisible processes and Levy processes generalize the Wiener processes to situations where increments can have non-Gaussian distributions. For a nice summary see these lecture slides
The origins of word Martingale is described in this paper by Mansuy
Nyquist paper from 1928 explaining the $4 k_{B} R T B$ formula