Project 4 Description (2026–2027)

Ergodicity of Stochastic Systems

Supervisor: Huaizhong Zhao
Research Area: Probability/ Stochastic Analysis

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Background

Ergodic theory is a deep and influential branch of mathematics with applications across many scientific fields. At its core, it studies the indecomposability of measure-preserving dynamical systems defined on measure spaces.

A central result in this area is Birkhoff’s Ergodic Theorem, which establishes that ergodicity is equivalent to a form of the law of large numbers: over long time periods, time averages converge to space averages. This property is also linked to the simplicity of the eigenvalue 1 of the associated transformation operator.

Ergodicity is a fundamental feature of many stochastic systems. The presence of stochastic noise can enhance mixing properties and improve the long-term stability of dynamical systems. Because stochastic systems naturally exhibit spreading and irreducibility, ergodicity arises as a common and important phenomenon. A simple example is a finite, irreducible Markov chain, where the ergodic theorem guarantees that the long-run proportion of time spent in any given state converges to its stationary probability.

More broadly, ergodicity plays a key role in understanding the long-term evolution of probability distributions in stochastic systems. It is closely related to concepts such as irreducibility, mixing, and the dispersion of mass or information.

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Project Overview

In this project, students will first study the measure-theoretic formulation of ergodicity for measure-preserving dynamical systems on probability spaces. Building on this foundation, they may then explore one of the following topics:

• Brownian motion on Wiener space, including the ergodicity of the shift map.
• Continuous-time Markov processes, where one studies transition probabilities and the Chapman–Kolmogorov equation. These processes can be lifted to canonical measure-preserving dynamical systems on path space.
• Generalised notions of ergodicity, including:
• Ergodicity of periodic measures, which arise in systems with time-periodic structure
• Ergodicity of quasi-periodic measures, capturing more complex recurrent behaviours
• Ergodicity under sublinear expectations, relevant in settings with model uncertainty and nonlinear probability frameworks

The project may further develop into the study of ergodicity in stochastic differential equations (SDEs), SPDEs or McKean-Vlasov equations, focusing on invariant or periodic measures and geometric ergodicity. Possible approaches include:

1. Probabilistic methods
• Strong Feller property
• Irreducibility arguments
2. Analytical methods
• Study of Fokker–Planck equations
3. Spectral methods
• Analysis of differential operator spectra

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Scope

Master’s students are expected to focus on one of the above aspects in depth. The topic lies within an active and evolving research area, with significant opportunities for further study beyond the scope of a Master’s project.

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Mode of Operation and Evidence of Learning

The project will be based on independent study supported by regular meetings with the supervisor. Emphasis will be placed on developing a deep conceptual understanding of the theory and its mathematical structure.

Students will engage with advanced literature, work through key results, and explore examples where appropriate. Evidence of learning will include:

• A written project report demonstrating clear understanding and mathematical rigour
• Well-structured exposition and appropriate use of mathematical notation
• An oral examination discussing the project and demonstrating understanding

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Prerequisites

Probability II and Mathematical Finance III would be suitable prerequisites. This project would be particularly suitable for students also intending to take the course Stochastic Analysis IV.

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References

• L. Arnold, Random Dynamical Systems, Springer-Verlag, Berlin, 1998.
• G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, Cambridge, 1996.
• C. Feng and H. Zhao, Random Periodic Processes, Periodic Measures and Ergodicity, Journal of Differential Equations, 2012.
• C. Feng and H. Zhao, Ergodicity of Sublinear Markovian Semigroups, SIAM Journal on Mathematical Analysis, 53(5), 5646–5681, 2021.
• M. Hairer, Ergodic Properties of Markov Processes, lecture notes, 2006 (available online).
• P. Walters, An Introduction to Ergodic Theory, Springer-Verlag, New York, 1982.