Lecture on Dirichlet forms and stochastic calculus

1. Functional analytic background

• 1.1 Resolvents, semigroups, generators

• 1.2 Coercive bilinear forms

• 1.3 Closability

• 1.4 Contraction properties

2. Examples

• 2.1 Classical energy forms

• 2.2 Hamza type condition, other closability results and Beurling-Deny formula

• 2.3 Other non-symmetric cases

3. Analytic potential theory of Dirichlet forms

• 3.1 Excessive, coexcessive, 1-reduced and 1-coreduced functions

• 3.2 ε-exceptional sets and φ-capacity

• 3.3 ε-quasi-continuity

• 3.4 Quasi-regularity

• 3.5 Measures associated to coexcessive functions

• 3.6 Some relations to probabilistic potential theory

4. Stochastic analysis by additive functionals

• 4.1 Positive continuous additive functionals and Revuz measure

• 4.2 Fukushima's decomposition of AF's

• 4.3 The local property

• 4.4 An example

• 4.5 The Lyons-Zheng decomposition

Appendix

• A1. Basics about the Bochner integral

• A2. Two theorems from functional analysis