This page gives a rough summary and lecture notes for each lecture. Sections from "Brownian Motion. An Introduction to Stochastic Processes" (De Gruyter, 2nd edition) by R.L. Schilling and L. Partzsch are indicated as (X.Y).

You should read Chapter 3 of the book on your own. Various constructions of the Brownian Motion are presented there. See also my notes on the construction by Lévy (which is covered in MATH 895).

WEEK 1

08/01 - Brownian motion (1), Gaussian processes (2.1) - Francesco Cellarosi

10/01 - The Brownian motion is a Gaussian process (2.1), σ-algebra generated by a process (2.2) - Francesco Cellarosi

WEEK 2

15/01 - The d-dimensional Brownian motion (2.2) - Francesco Cellarosi

17/01 - Invariance properties of Brownian motion (2.3) - Bora Yongacoglu

WEEK 3

22/01 - The Wiener measure (4.1) - Daniel Adu

24/01- Kolmogorov's "construction" (4.2) - Tariq Osman

WEEK 4

29/01 - Brownian Martingales (5.1) - Neil MacVicar

31/01 - Stopping and Sampling (5.2) - Curtis McDonald

WEEK 5

05/02 - Markov Property (6.1) - Somya Singh

07/02 - Strong Markov Property (6.2) - Ankai Liu

WEEK 6

12/02 - CLASS CANCELLED

14/02 - Reflection Principle (6.3) - Ali Kara

WEEK 7

26/02 - Transience and Recurrence (6.4) - Bora Yongacoglu

28/02 - Some Measurability Issues (6.7) - Francesco Cellarosi

WEEK 8

05/03 - Transition Semigroup (7.1) - Francesco Cellarosi

07/03 - Generator (7.2) - Curtis McDonald

WEEK 9

12/03 - Potential operator and Resolvent (7.3) - Nicolas Garcia

14/03 - Hille-Yosida Theorem and positivity (7.4) - Nicolas Garcia

WEEK 10

19/03 - More on the generator and the potential. How to solve 𝛼u-Au=f (7.3-7.5) - Francesco Cellarosi

21/03 - Solving the Heat Equation (8.1) - Neil MacVicar

WEEK 11

26/03 - Inhomogeneous Initial Value Problems (8.2) - Somya Singh

28/03 - Feynman-Kac Formula (8.3) - Tariq Osman

WEEK 12

03/04 - Dirichlet Problem (8.4) - Daniel Adu

04/04 - Dirichlet Problem (8.4) - Ali Kara