Topics in the Theory of Markov Processes

What it is about

You engage with analytical aspects of Markov processes, in particular Itô diffusions (linear and nonlinear). By the end you will be able to understand Markov semigroups and generators, classify basic existence and uniqueness results for McKean-SDEs/McKean–Vlasov equations, and make statements about long-term behavior as well as phase transitions for nonlinear Fokker–Planck equations.

What you will be able to do

• Understanding of Markov semigroups and their infinitesimal generators
• Knowledge of ergodic theory and convergence to equilibrium for Itô diffusions
• Familiarity with functional inequalities and Bakry-Émery/Gamma calculus
• Derivation of McKean-SDE and McKean–Vlasov equation as mean-field limit
• Fundamental existence and uniqueness statements for McKean-SDEs and McKean–Vlasov equations
• Assessment of long-term behavior, non-uniqueness of invariant measures and relation to phase transitions

What the module consists of

• LectureConveying theoretical content with examples and discussion; central teaching event
• Exercises/HomeworkDeepening the material through two assignments
• Project workWritten report and presentation on application or deepening

Teaching method

• Lecture/Instructor Talkfor systematic presentation of the content
• Discussion with studentsfor deepening and critical engagement
• Demonstrative examples and exercisesfor illustration and independent analysis by students
• Project work with report and presentationfor independent application and presentation of results