# TCC/CO905: Large-Scale Dynamics of Stochastic Particle Systems

The module is part of the Mathematics Taught Course Centre.

**Lectures: Thursdays 3-5pm in the TCC room (B0.06)**

**First lecture: 19.1. (week 2)**

**Regularly updated course notes (Chapters updated up to 4.4): **notes_tcc.pdf

**Assessment.** For Warwick MSc students this module can be taken for credit under the code CO905, for which additional suppoert classes are available per week, and which is assessed by essay and a short presentation. Assessment for PhD students can also be made available if necessary.

More information on essay topics and specifications can be found here.

**Content.** Stochastic particle systems are probabilistic models of complex phenomena that involve a large number of interacting components, such as the spread of epidemics or traffic flow. The module will focus on a mathematical description of simple minimal models and their large scale dynamic properties and fluctuations. There has been significant recent research activity in this area leading to the formulation of a comprehensive approach called 'Macroscopic Fluctuation Theory', an introductory description of which is the main goal of this module.

**Syllabus**

- Introduction and basic review of continuous-time Markov chains and diffusion processes
- Characterization of processes in terms of path measures, generators, semigroups and martingale problems

- Stochastic particle systems, graphical construction and examples, including exclusion, zero-range and contact process

- stationary measures and time reversal, current, ergodic properties, conservation laws and symmetries

- Useful analytical tools, including Girsanov formula, Feynman-Kac formula, Doob h-transform and extensions

- hydrodynamic limits and fluctuating hydrodynamics

- fluctuation relations, Gallavotti-Cohen theorem, entropy production

- macroscopic fluctuation theory and applications to several examples

**Prerequisites.** A background in continuous-time Markov chains is necessary, background in diffusion processes (or Brownian motion) and measure theory is very useful but not strictly necessary, further background in functional analysis and basic PDE theory can be useful.

**Literature.** Typed course notes will be made available.

- L. Bertini, A. Faggionato, D. Gabrielli: Large deviations of the empirical flow for continuous time Markov chains. Ann. Inst. H. Poincare Volume 51, 867, 2015 (arXiv:1210.2004)

- L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim: Macroscopic Fluctuation Theory. Reviews of Modern Physics, Volume 87, 593-636, 2015 (arXiv:1404.6466)

- R. Chetrite, H. Touchette: Nonequilibrium Markov processes conditioned on large deviations. Ann. Henri Poincare 16, 2005, 2015 (arXiv:1405.5157)

- Chapters 3 and 4 of: T.M. Liggett: Continuous Time Markov Processes. AMS Graduate Studes in Mathematics 113, 2010
- C. Kipnis, C. Landim: Scaling Limits of Interacting Particle Systems. Springer, 1999