Content: In 1827 the Botanist Robert Brown reported that pollen suspended in water exhibit random erratic movement. This ‘physical’ Brownian motion can be understood via the kinetic theory of heat as a result of collisions with molecules due to thermal motion. The phenomenon has later been related in Physics to the diffusion equation, which led Albert Einstein in 1905 to postulate certain properties for the motion of an idealized ‘Brownian particle’ with vanishing mass:
- the path $t\mapsto B(t)$ of the particle should be continuous,
- the displacements $B(t+\Delta t)-B(t)$ should be independent of the past motion, and have a Gaussian distribution with mean 0 and variance proportional to $\Delta t$.
In 1923 ‘mathematical’ Brownian motion was introduced by the Mathematician Norbert Wiener, who showed how to construct a random function $B(t)$ with those properties. This mathematical object (also called the Wiener process) is the subject of this module.
Over the last century, Brownian motion has turned out to be a very versatile tool for theory and applications with interesting connections to various areas of mathematics, including harmonic analysis, solutions to PDEs and fractals. It is also the main building block for the theory of stochastic calculus (see MA482 in term 2), and has played an important role in the development of financial mathematics. Even though it is almost 100 years old, Brownian motion lies at the heart of deep links between probability theory and analysis, leading to new discoveries still today.
Topics discussed in this module include:
- Construction of Brownian motion/Wiener process
- fractal properties of the path, which is continuous but still a rough, non-smooth function
- connection to the Dirichlet problem, harmonic functions and PDEs
- the martingale property of Brownian motion and some aspects of stochastic calculus
- description in terms of generators and semigroups
- description as a Gaussian process, an important class of models in machine learning
- some generalizations, including sticky Brownian motion and local times
Peter Mörters and Yuval Peres, Brownian Motion, Cambridge University Press, 2010
Thomas M. Liggett, Continuous Time Markov Processes - An Introduction, AMS Graduate studies in Mathematics 113, 2010