Special lectures on advanced topics in Mathematics E

Academic unit or major

Graduate major in Mathematics

Instructor(s)

Yushi Hamaguchi / Masato Hoshino

Class Format

Lecture (Face-to-face)

Media-enhanced courses

-

Day of week/Period
(Classrooms)

Intensive (Mathematics Seminar Room 201)

Class

-

Course Code

MTH.E435

Number of credits

200

Course offered

2026

Offered quarter

2Q

Syllabus updated

Mar 30, 2026

Language

Japanese

Syllabus

Course overview and goals

The primary theme of this course is the ergodicity of Markov processes in infinite-dimensional spaces and its application to the Markovian lifting of stochastic Volterra equations (SVEs). As an extension of stochastic differential equations, SVEs have attracted significant attention as models for describing non-Markovian phenomena with "memory effects" in fields such as mathematical finance and statistical physics. However, their non-Markovian nature poses substantial challenges for the direct application of classical Markovian techniques to analyze their long-time behavior. To address this, the "Markovian lifting" method has recently been developed, which reformulates the solution of an SVE as a Markov process in an infinite-dimensional Hilbert space. This course provides a systematic study of the general theory of ergodicity for infinite-dimensional Markov processes and its concrete application to the Markovian lifts of SVEs.
As a powerful general theory for the ergodicity of infinite-dimensional Markov processes, we will provide a detailed exposition of a generalized form of Harris’ theorem established by Hairer–Mattingly–Scheutzow (2011). This theorem is an extremely versatile tool, applicable not only to SVEs but also to stochastic delay differential equations and certain classes of stochastic partial differential equations. In the latter half of the course, we will introduce the framework for the Markovian lifts of SVEs. By applying the aforementioned generalized Harris’ theorem, we will present recent research results on deriving the unique existence of invariant probability measures and the weak convergence of transition probabilities (weak ergodicity).

Course description and aims

・To understand the statement of the generalized version of Harris’ theorem.
・To grasp the concept of Markovian lifting for stochastic Volterra equations.
・To understand the proof of ergodicity for the Markovian lifts of stochastic Volterra equations.

Keywords

Markov process, invariant probability measure, ergodicity, Harris’ theorem, generalized coupling, stochastic Volterra equation, Markovian lift

Competencies

• Specialist skills
• Intercultural skills
• Communication skills
• Critical thinking skills
• Practical and/or problem-solving skills

Class flow

This is a standard lecture course. There will be some assignments.

Course schedule/Objectives

Study advice (preparation and review)

Textbook(s)

None required.

Reference books, course materials, etc.

・M. Hairer, J.C. Mattingly, and M. Scheutzow. Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149, 223--259, 2011.
・Y. Hamaguchi. Markovian lifting and asymptotic log-Harnack inequality for stochastic Volterra integral equations, Stochastic Process. Appl., 178, 104482, 2024.
・A. Kulik. Ergodic Behavior of Markov Processes: With Applications to Limit Theorems, Berlin, Boston: De Gruyter, 2018. https://doi.org/10.1515/9783110458930

Evaluation methods and criteria

Assignments (100%).

Related courses

• MTH.C361 ： Probability Theory
• MTH.C507 ： Advanced topics in Analysis G1
• MTH.C508 ： Advanced topics in Analysis H1

Prerequisites

None