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# M1 Brownian Motion and Stochastic Processes

## Archives

Exercise sessions for the lecture given by Hélène Leman at ENS Lyon in the Spring of 2020.

- Exercise sheet #1 - solutions
- Exercise sheet #2 - solutions
- Exercise sheet #3 - solutions
- Exercise sheet #4-5- solutions
- Exercise sheet #6- solutions

In 2018 and 2019, the courses were given by Emmanuel Jacob. A lot of material originates from my predecessors, Sébastien Martineau and Xiaolin Zeng, and from the book *Brownian Motion* by Peter Mörters and Yuval Peres. Here are the archives from 2019.

2020 students: spoiler alert, avoid looking at solutions here too much in advance

- Exercise sheet #1: Conditional distributions, first properties of Brownian motion - solutions
- Exercise sheet #2: Properties and construction of Brownian motion - solutions
- Exercise sheet #3: Stopping times and Markov property - solutions (to be completed)
- Exercise sheet #4: Markov processes, regularity of Brownian motion - solutions
- Exercise sheet #5: Martingales - solutions
- Exercise sheet #6: Some more martingales, Donsker's invariance principle - solutions
- Exercise sheet #7: Donsker's invariance principle - solutions
- Exercise sheet #8: Miscellaneous, Azéma-Yor embedding - solutions
- Exercise sheet #9: Harmonic functions and Brownian motion - solutions
- Exercise sheet #10: Brownian motion, harmonic functions and measures - solutions
- Exercise sheet #11: Haussdorf dimension, Poisson equation - solutions
- Some more exercises from 2018
- 2019 Homework assignment: On the Brownian bridge - solutions
- 2019 Midterm - solutions (in French)
- 2019 Final exam - solutions (in French)
- 2018 Homework assignment: Hausdorff dimension of the zero set of Brownian motion - solutions
- 2018 Midterm - solutions
- 2018 Final exam

For the sake of history, here are the complete archives of 2018.