We covered the material contained in the lecture notes of Perla Sousi on pages 1-70 (without theorem 2.34, and with theorem 6.26 proved only for a harmonic function f(x)). In the proof of 6.29, we additionally stated a geometric lemma which can be found in P. Mörters' and Y. Peres' book as Lemma 3.11.

Additional sources are the lecture notes of James Norris and Grégory Miermont, and books:

1. D. Williams, Probability with Martingales, Cambridge University Press 1991 (Part B)
2. P. Mörters and Y. Peres, Brownian motion, Chapters 1,2,3,5, Cambridge University Press 2010
3. R. Durrett, Probability: Theory and Examples, Wadsworth 1991
4. L.C.G. Rogers and D. Williams, Diffusions, Markov processes and Martingales Vol. I (2nd edition), Chapters I & II, Wiley 1994

The Examples classes are on the 1st, 15th and 29th of November, and on the 24th of January, at 14:00 in MR 9

Problems to be handed in (before the last preceding Friday, at 17:00, pigeonhole W/X) :

• Example sheet 1: Q1.3, Q2.3
• Example sheet 2: Q3.8, Q3.10
• Example sheet 3: Q6.1, Q7.4, Q5.3 (choose 2 or 3)
• Example sheet 4 (only the first page): Q7.10 and a proof of the upper bound in the law of the iterated logarithm for Brownian motion.

The revision Examples class will be held in MR9 on 18th May from 12pm to 2pm. The plan is to go through some of the questions from the past exam papers 2015-2017. People are encouraged to submit their solutions to any of the two questions (but just 2 - if one submits more than 2, only 2 will be picked randomly for marking) from the list below:

2017: Q1, 2, 6

2016: Q1, 2, 3, 4, 5

2015: Q1, 2, 3, 5.

You should submit your work to the pigeonhole of Mo Dick Wong (CCA pigeonhole, W/X) by 12pm on 16th May.