C8.2 Stochastic Analysis and PDEs - Material for the year 2019-2020

2019-2020
Lecturer(s): 
Ilya Chevyrev
General Prerequisites: 

Integration and measure theory, martingales in discrete and continuous time, stochastic calculus. Functional analysis is useful but not essential.

Course Term: 
Hilary
Course Lecture Information: 

16 lectures

Course Weight: 
1.00 unit(s)
Course Level: 
M

Assessment type:

Course Overview: 

Stochastic analysis and partial differential equations are intricately connected. This is exemplified by the celebrated deep connections between Brownian motion and the classical heat equation, but this is only a very special case of a general phenomenon. We explore some of these connections, illustrating the benefits to both analysis and probability.

Learning Outcomes: 

The student will have developed an understanding of the deep connections between concepts from probability theory, especially diffusion processes and their transition semigroups, and partial differential equations.

Course Synopsis: 

Feller processes and semigroups. Resolvents and generators. Hille-Yosida Theorem (without proof). Diffusions and elliptic operators, convergence and approximation. Stochastic differential equations and martingale problems. Duality. Speed and scale for one dimensional diffusions.
Green's functions as occupation densities. The Dirichlet and Poisson problems. Feynman-Kac formula.

Reading List: 

A full set of typed notes will be supplied.

Important references:

  1. O. Kallenberg. Foundations of Modern Probability. Second Edition, Springer 2002. This comprehensive text covers essentially the entire course, and much more, but should be supplemented with other references in order to develop experience of more examples.
  2. L.C.G Rogers & D. Williams. Diffusions, Markov Processes and Martingales; Volume 1, Foundations and Volume 2, Itô calculus. Cambridge University Press, 1987 and 1994. These two volumes have a very different style to Kallenberg and complement it nicely. Again they cover much more material than this course.
Further Reading: 
  1. S.N. Ethier & T.G. Kurtz. Markov Processes: characterization and convergence. Wiley 1986. It is not recommended to try to sit down and read this book cover to cover, but it is a treasure trove of powerful theory and elegant examples.
  2. S. Karlin & H.M. Taylor. A second course in stochastic processes. Academic Press 1981. This classic text does not cover the material on semigroups and martingale problems that we shall develop, but it is a very accessible source of examples of diffusions and things one might calculate for them.

A fuller list of references will be included in the typed notes.