Cours 1 : Définitions de base, MC comme systèmes dynamiques aléatoires, Exemples de MC (marches aléatoires). Discussion rapide du cas inhomogène. La propriété de Markov (simple). Temps d'arrêt et propriété de Markov forte (à completer). Jusqu'à page 11 (Ch.1)

Cours 2 : La propriété de Markov forte. Noyau potentiel : Proposition 5.2 et 5.3. Etats récurrents : Prop. 5.4. Mesures invariantes et mesures réversibles (Ch. 1, Sec. 6). Espace d'états dénombrable (Ch., Sec. 7). Exemple : la marche aléatoire simple (Sec. 7.2), à completer.

Cours 3 : Exemple : la marche aléatoire simple (Sec. 7.2) et le processus de naissance et mort (sec. 7.3). Ch. 2 : états accessibles et récurrents. Le processus de Lindley (introduction). La mesure $\lambda_x$. Th. 2.1 : énoncé complet. TH. 2.1 (1) pas loin d'être démontré.

Notes (24/09/2023)} révisées jusqu'à page 36, à l'exception des sections 7.3 est 7.4 du ch. 1.

Notes (17/09/2023 : plusieurs corrections locales, jusqu'à la section 7 du Ch. 1 (partiellement révisée).

Notes (01/09/2023)

The course starts on Wednesday (9:00-12:00) February 8th in building Lamarck B, room 204. The rest of the classe will be in Building Lavoisier, room 227, except for the very last class that in principle will be again in Lamarck B, room 204 (but we will see).

An important issue is how to get to these two classrooms. First of all you have to get to a neighborhood of the Sophie Germain building (but the classes are not in the Sophie Germain building!) and for this you can either use M14, bus 89, RER C, etc… In any case if you use the RER C exit (see here for a map) or if you get off from bus 89 you are 50~100 m. from Sophie Germain

• For Lamarck B 204, go from Avenue de France down rue Brion and enter the building to your right (it is the second entry but normally it is the only one for which the grid is open). Take one of the escalators (or the elevator) on your right and go to up two floors. Follow the arrows with the room numbers.
• For Lavoisier 227 go to the entry grid of Sophie Germain. Do not enter and avoid the building on the left (going around on the right is longer, but you can do that if you prefer). Take a right just behind the building, on rue Einstein (on the map it is written Chauvin, but it is Einstein), and go to the entry of the Olympe de Gouges building. Enter the grid of Olympe de Gouges and keep as left as you can. Go down a flight of stairs: the building on the right is Lavoisier and if you can enter you are on the 2nd floor and 227 is there. If that door is closed, go down another flight of stairs and take the main entry.

Here are the lecture notes I will use.

Here are the updated lecture notes (April 5th): I am going to make some corrections (and possibly I will add explanations etc…). I will in any case list here the changes (except possibly for very minimal corrections).

List of changes:

1. p. 16, paragraph after (1.43): added some details
2. p. 17, line before (1.45): $X_n$ replaced by $M_n$
3. p. 18, two lines before (1.51): reference to Lemma 3.15 replaced by reference to Lemma 1.13
4. p. 18, paragraph of (1.51): corrected non compactness argument (and $|det(M)|=1$ taken off from (1.51): it was superfluous)
5. p. 52, some notational changes
6. p. 53, corrected sign in (3.34) and in last line of (3.35)
7. p. 56, corrected exponent 2 to -2 in the proof on top of the page

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