Introductory description

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↦B(t) of the particle should be continuous,
• the displacements B(t+Δt)−B(t) B(t+Δt)−B(t) should be independent of the past motion, and have a Gaussian distribution with mean 0 and variance proportional to Δt

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

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
• description as a Markov process in terms of generators and semigroups
• scaling properties and connection to random walk
• some generalizations, including e.g. geometric Brownian motion and fractional Brownian motion

Learning outcomes

By the end of the module, students should be able to:

• - describe its construction and explain simple properties of Brownian Motion (BM);
• - understand BM as a continuous time and continuous state Markov process;
• - use the martingale property of BM to derive advanced properties such as Wald’s lemmas;
• understand the embedding of random walks in Brownian motion and use it to derive convergence results;
• - translate properties of one-dimensional BM to higher dimensions.

Subject specific skills

At the end of the module students will be able to :

• describe its construction and explain simple properties of Brownian Motion (BM);
• understand BM as a continuous time and continuous state Markov process;
• use the martingale property of BM to derive advanced properties such as Wald’s lemmas;
• understand the embedding of random walks in Brownian motion and use it to derive convergence results;
• translate properties of one-dimensional BM to higher dimensions.

Transferable skills

Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.