Stochastic processes and complex systems

This course provides an introduction to stochastic processes and their role in the study of complex systems. Random processes arise in many areas of contemporary science, ranging from Brownian motion and statistical physics to population dynamics, epidemiology, quantitative finance, and engineering.

The course introduces the principal levels of description of stochastic processes through a series of simple yet fundamental models. Starting from examples of Markov chains and urn models, we progressively develop the Kolmogorov master equation, the Fokker-Planck equation, and the Langevin equation, which form the backbone of modern stochastic-process theory.

We then study the fundamental concepts of Brownian motion, white noise, and Wiener processes, which naturally lead to stochastic integration and the Itô and Stratonovich formalisms.

Throughout the course, particular emphasis is placed on the connections between microscopic and macroscopic descriptions, as well as on the ways in which complex collective behaviour can emerge from simple random mechanisms.

Stochastic processes play a central role in many areas of contemporary science, notably in statistical physics, but also in biology, chemistry, engineering, and the study of complex systems.

The objective of this course is to provide students with the fundamental conceptual and mathematical tools of stochastic process theory, together with an understanding of the connections between different descriptions of random phenomena, ranging from microscopic dynamics to continuous evolution equations.

Upon completion of the course, students will be able to analyze and model simple stochastic systems, work with the main tools of the discipline (Markov chains, master equations, Fokker-Planck and Langevin equations, stochastic calculus), and engage independently with the scientific literature on the subject.

Required and Corequired knowledge and skills

The course assumes a basic knowledge of calculus, linear algebra, probability theory, and statistical physics. Familiarity with scientific programming (Python or any comparable programming language) is recommended.

The course consists of twelve two-hour lectures. Beyond introducing the fundamental concepts of stochastic-process theory, the course emphasizes the connections between different levels of description of random systems, as well as more recent developments inspired by contemporary research.

Exercises and problem sets are assigned regularly to promote an active understanding of the material and to encourage students to develop their own intuition for stochastic phenomena. Submitted solutions contribute to the final assessment.

Lecture notes, recommended readings, problem sets, and other learning resources are made available through the Virtual University (UV) platform.

References, bibliography, and recommended reading

• C W Gardiner, Stochastic Methods (Springer 2009)
• N G van Kampen, Stochastic Processes in Physics and Chemistry (Elsevier 2007)
• W Horsthemke et R Lefever, Noise-Induced Transitions (Springer 1984)

Course notes

• Université virtuelle

Contacts

thomas.gilbert@ulb.be
Campus Plaine, bâtiment NO, 5e étage, bureau P.2.O5.105

Campus

Plaine

Method(s) of evaluation

• Continuous assessment
• Oral examination

Continuous assessment

Oral examination

• Examination with preparation

• Weekly homework assignments, submitted throughout the semester and accounting for 50% of the final grade.

• Final oral examination, including a written preparation phase followed by a discussion, and accounting for the remaining 50% of the final grade.

Mark calculation method (including weighting of intermediary marks)

Evenly distributed between homework grading and the final exam

Language(s) of evaluation

• english
• (if applicable french )