Project 3 Description (2026–2027)

Periodic Measures in Stochastic Systems

Supervisor: Huaizhong Zhao
Research Area: Probability / Stochastic Analysis

Background

Many real-world systems exhibit time-periodic behaviour, such as seasonal effects in climate models, biological rhythms, and oscillatory physical systems. While periodicity in deterministic systems is well understood through periodic orbits, stochastic systems involve additional complexity due to the interaction between randomness and periodic forcing.

A natural framework for studying such systems is provided by the concept of a periodic measure, which generalises the notion of an invariant measure to time-dependent settings. Unlike invariant measures, which remain constant over time, periodic measures evolve cyclically and reflect the underlying temporal structure of the system.

Periodic measures offer a powerful tool for describing the long-term behaviour of stochastic systems subject to periodic inputs. They also form an important link between classical ergodic theory and modern developments in stochastic dynamics.

Group Project

The group project introduces the basic theory of stochastic processes with time-periodic structure, with a focus on understanding periodic measures and their properties.

Students will begin with:

A review of Markov processes and invariant measures
Fundamental concepts of ergodicity and long-term behaviour
An introduction to time-periodic stochastic systems

The project will then develop the theory of periodic measures, including:

Their definition and interpretation
Their relationship to invariant measures and random periodic paths
Illustrative examples arising from simple stochastic differential equations (SDEs)

Mode of Operation and Evidence of Learning

The project is based on guided reading and independent study, with emphasis on developing a solid theoretical understanding and mathematical rigour.

Students will demonstrate their learning through:

Solving relevant problems
Exploring examples and applications
Presenting their findings clearly in written reports and oral presentations

Individual Project: Possible Directions

Following the group component, students may pursue an individual topic in more depth. Possible directions include:

Stochastic resonance
Investigating how noise can enhance periodic signals in dynamical systems, and how periodic measures describe this behaviour
Large deviations for periodic systems
Developing an introductory understanding of rare events in periodically forced stochastic systems
Numerical methods
Studying computational approaches for approximating periodic measures and simulating stochastic systems
Ergodic optimal control (introductory level)
Exploring long-term average behaviour in controlled stochastic systems with periodic structure

These topics will be approached at an accessible and conceptual level, with emphasis on intuition, examples, and simple calculations rather than technical proofs.

Prerequisites

Probability II is an essential prerequisite. This project would be suitable for students also intending to take the course Mathematical Finance III. Taking Stochastic Processes III alongside may also be helpful.

References

H. Zhao and Z. Zheng, Random Periodic Solutions of Random Dynamical Systems, Journal of Differential Equations, 246(5), 2009, 2020–2038.
C. Feng and H. Zhao, Random Periodic Processes, Periodic Measures and Ergodicity, Journal of Differential Equations, 269(9), 2020, 7382–7428.
R. Bai, C. Feng, and H. Zhao, Large Deviation Principle for Invariant Measures of Stochastic Burgers Equations, Journal of Functional Analysis, 290, 2026, Article 111284.
C. Feng, Y. Liu, Y. Liu, and H. Zhao, Ergodic Semi-Implicit Approximations to Periodic Measures of Stochastic Differential Equations with Locally Lipschitz Drifts: Error Analysis in Wasserstein Distance, Journal of Differential Equations, 441, 2025, Article 113472.