ST403-15 Brownian Motion

Introductory description

This module runs in Term 1 and is only available for students with their home department in Statistics. It is delivered by the Mathematics Department under the module code MA4F7.

Prerequisites: ST318 Probability Theory OR MA359 Measure Theory.

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(s+t)-B(s) should be independent of the past motion, and have a Gaussian distribution with mean 0 and variance proportional to t

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
• description as a Gaussian process, an important class of models in machine learning
• description as a Markov process in terms of generators and semigroups
• the martingale property of Brownian motion and some aspects of stochastic calculus
• scaling properties and connection to random walk
• connection to the Dirichlet problem, harmonic functions and PDEs
• 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:

• Use the martingale property of BM to derive advanced properties such as Wald's lemmas;
• Describe its construction and explain simple properties of Brownian Motion (BM);
• Understand BM as a continuous time and continuous state Markov process;
• 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.

Indicative reading list

Peter Mörters and Yuval Peres, Brownian Motion, Cambridge University Press, 2010
René L. Schilling and Lothar Partzsch, Brownian motion: an introduction to stochastic processes, De Gruyter, 2014
Thomas M. Liggett, Continuous Time Markov Processes - An Introduction, AMS Graduate studies in Mathematics 113, 2010

View reading list on Talis Aspire

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.