Processus stochastiques et systèmes complexes

The notion of stochastic process is very broad. In this course, it will be introduced through the example of an urn model, the evolution of which is specified by probabilistic laws. The statistical study of this model makes it possible to illustrate the levels of description which form the backbone of the theory of stochastic processes:

• the Kolmogorov equation (or master equation),
• the Fokker-Planck equation,
• Langevin's equation.

We will also see how such a model of urns can emerge from the deterministic dynamics of a mechanical system. We will then develop fundamental concepts such as Brownian motion and white noise, or Wiener process. This will naturally lead to the notion of stochastic integration and to the two conventions of Itô and Stratonovich. Different models will be introduced in the sequel, which allow to deepen the basic notions and the links between different levels of description.

Stochastic processes occupy a central role in many areas of contemporary science, particularly in physics and more specifically in statistical physics. The objective of this course is to set up the fundamental tools of the theory of stochastic processes, while emphasising their relevance in the context of recent developments. At the end of the training, students will be able to continue their learning of the subject on their own and make their own contribution to the field.

Required and Corequired knowledge and skills

• Calculus
• Statistical physics

The lectures are divided up into 12 sessions of 2 hours each. Informal discussions will be held on a weekly basis during which students are encouraged to make their own contributions. Exercises will also be offered. The course material (bibliographic sources or lecture notes) is made available via the virtual university.

Contribution to the teaching profile

• Acquire scientific expertise in the field of physics
• Master the scientific process
• Communicate in a language adapted to the context and its audience

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

• Oral examination
• Oral presentation
• Other

Oral examination

Oral presentation

Other

Mark calculation method (including weighting of intermediary marks)

Evenly distributed among the different assessment methods

Language(s) of evaluation

• french
• (if applicable english )