Title: MATH 663 Dirichlet Forms, Markov Processes and Semigroups

Prerequisites: MATH 607 (Real Analysis) and MATH 411 (Undergraduate Probability). Useful but not required: MATH 606 (Probability)

Course Description: The course will be an introduction to the theory that relates the concepts of Dirichlet forms, semigroups and Markov processes. It lies at the
interface between functional analysis and probability and the theory is widely applied in the context of analysis and probability in metric measure spaces.
I. Semigroups, Markov processes and generators
- Semigroup theory
- Infinitesimal generators
- Resolvents and the hille-Yosida theorem
II. Dirichlet forms, Markov processes and electric networks
- Definitions
- Basic relationships
- Electric networks and compatible sequences
(III. Brownian motion and harmonic functions - time permitting)

Average time dedicated per week (estimate): Short expository writing (~2x in semester), optional homework (~3x in semester), final presentation.