# MA4F7 Content

Content: Brownian motion was originally the description given in physics for the random erratic movement of molecules. In 1905 Einstein made a detailed study in which he postulated certain properties should hold. In 1923 mathematical Brownian motion was born when a famous mathematician, Norbert Wiener, showed how to construct a random function $W(t)$ giving the molecules "position'' at time t which had Einstein's properties.

According to Einstein:

• the function $W(t)$ must be continuous - the molecule never jumps;
• the displacement between times s and t, that is $[W(t) - W(s)]$, should be independent of the past motion and its distribution should be Gaussian with mean zero and variance $t-s$.

We will investigate methods of constructing such random functions. It turns out the Gaussian distribution is essential - it is impossible to do with any other distribution.

• Properties of the paths. The path $W(t)$ cannot be smooth. Look at www.stat.umn.edu/~charlie/Stoch/brown.html to see a simulation. The applet at this web site allows you to zoom in on a simulated path - notice it seems to look the same no matter how much it is magnified: Brownian motion is the ultimate fractal!
• The stochastic calculus. Ordinary calculus is a powerful method of doing calculations with smooth functions. As we have just seen Brownian paths are not smooth, but miraculously there is a stochastic calculus'' which was developed by a Japanese mathematician Ito in the 1940s and which allows us to do computations with Brownian motion.
• Differential equations. Differential equations are essential to modelling deterministic phenomena in applied maths and physics. Arguably the most famous equation of all is Laplace's equation:

$$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} + \frac{\partial^2u}{\partial z^2} = 0$$

Somewhat surprisingly this can be solved probabilistically using Brownian motion - a fact that lies at the heart of the links between probability theory and analysis, and which is still today yielding new discoveries.