PILLAR Course: Probability and Statistical Mechanics - Part 2: Probability theory for statistical mechanics

Monday 2 Dec 2024, 08:30 → Wednesday 18 Dec 2024, 12:30 Europe/Rome

Lecturer

Lu Xu lu.xu@gssi.it 

 

Course content

1. Preliminary

1.1 Random variable

• probability space, probability distribution, probability density function, expectation, independence
• important examples: exponential distribution, Gaussian distribution, Gaussian vectors

1.2 Conditional expectation

• definition, basic properties
• computation of conditional expectation

1.3 Convergence

• strong and weak convergence
• vague topology, Prokhorov’s theorem

1.4 Basic measure theory*

• absolutely continuity, Radon–Nikod´ym theorem*
• Lebesgue’s decomposition theorem*

2. Limit theorems

2.1 law of large numbers

• topology on infinite product space, i.i.d. sequence
• strong law for i.i.d. sequence
• central limit theorem (i.i.d. case)
• large deviations (i.i.d. case)
• moment-generating function, Legendre transformation

3. Stochastic process

3.1 Basic concepts

• sample paths, distribution, finite dimensional distribution

3.2 Poisson process

• Poisson distribution, distribution of Poisson process
• semigroup, infinitesimal generator
• limit theorems, homogenization

3.4 Brownian motion

• construction of Brownian motion, Donsker’s theorem, Wiener measure
• Brownian sample paths*
• a first look at (Ito) stochastic integral*

4. Poisson point process*

4.1 Introduction to point process*

• definition of Poisson process as a point process*

4.2 Poisson point process on Rd*

• definition, law of large numbers, central limit theorem*

 

Timetable and workload

• Monday, Dec. 2nd, 8:30-10:30
• Tuesday, Dec. 3rd, 10:30-12:30
• Thursday, Dec. 5th, 8:30-10:30
• Tuesday. Dec. 10th, 10:30-12:30
• Wednesday, Dec. 11th, 10:30-12:30
• Thursday, Dec. 12th, 8:30-10:30
• Monday, Dec. 16th, 8:30-10:30
• Tuesday, Dec. 17th, 10:30-12:30
• Wednesday, Dec, 18th, 10:30-12:30

 

Lecture Notes

A draft of lecture notes can be found here.