Content
Stochastic processes are mathematical models used to describe the dynamics of random phenomena
and are widely applied in many disciplines ranging from physics, chemistry, biology, and economics.
During the course, students will learn both the theory underlying stochastic processes and advanced numerical
methods to solve problems with real applications.

Website of the course

Github repository which collects the jupyter notebooks and the notes presented in class. The notebooks can be opened and used in the browser by means of binder (no need to download/install anything).

Topics for the exam.

Lecture 13: Conclusion and overview of the course

• Overview

Lecture 12: Fuzzy clustering and PCCA+

• Lecture notes 12
• Lecture 12, jupyter notebooks

Suggested readings

• Deuflard P. and Weber M., 2004

Lecture 11: Square Root Approximation (SqRA) of the infinitesimal generator

• Lecture notes 11
• Lecture 11, jupyter notebooks

Suggested readings

• Donati L., 2018
• Donati L., 2021

Lecture 10: Transfer operator formalism

• Lecture notes 10
• Lecture 10, jupyter notebooks

Suggested readings

• Klus S. 2018

Lecture 9: Kramers rate theory for low friction regime

• Lecture notes 9 (Work in progress...)
• Lecture 9, jupyter notebooks

Suggested readings

• Baron P., Reaction Rate Theory and Rare Events Simulations, chapter 16
• Kramers H.A. 1940

Lecture 8: Kramers rate theory for moderate and high friction regime

• Lecture notes 8 (Work in progress...)
• Lecture 8, jupyter notebooks
• Exercise 8

Suggested readings

• Baron P., Reaction Rate Theory and Rare Events Simulations, chapter 16
• Kramers H.A. 1940

Lecture 7: Introduction to escape rate problem, backward Kolmogorov equation, Mean First Passage Time, Pontryagin's formula, Ornstein_Uhlenbeck process, integration schemes for SDEs

• Lecture notes 7
• Lecture 7, jupyter notebooks
• Exercise 7
• Exercise 7, jupyter notebooks

Suggested readings

• Baron P., Reaction Rate Theory and Rare Events Simulations, chapter 18

Lecture 6: Fluctuation-Dissipation Theorem; overview of Fourier analysis; from Generalized Langevin Equation to Langevin Dynamics; introduction to Stochastic Calculus; System Size expansion method for Master equations

• Lecture notes 6
• Lecture 6, jupyter notebooks

Suggested readings

• Kubo R. 1966
• Van Kampen N.G., The Expansion of the Master Equation 1976, chapters 1-5

Lecture 5: Overview of Hamiltonian dynamics and Statistical Mechanics; The Generalized Langevin Equation: the memory kernel and the noise term

• Lecture notes 5
• Notes Hamiltonian Dynamics
• Notes Statistical Mechanics
• Notes integrators
• Lecture 5, jupyter notebooks

Suggested readings

• Kupferman R. 2002

Lecture 4: Derivation of the Generalized Langevin Equation from the Kac-Zwanzig model (4a); method of generating function and Gillespie's algorithm to solve the master equation (4b)

• Lecture notes 4a
• Lecture notes 4b
• Lecture 4, jupyter notebooks
• Exercise 4
• Exercise 4, jupyter notebooks

Suggested readings

• Tuckerman M.E., Statistical Mechanincs: Theory and Molecular Simulation 2010, Chapter 15
• Gillespie D.T., Markov Processes 1992, Chapter 5
• Shorack G.R., Probability for statisticians 2000, Chapter 7

Lecture 3: Markov processes, derivation of Chapman-Kolmogorv equation, Kramers-Moyal expansion, master equation, Fokker-Planck equation, Pawula theorem

• Lecture notes 3
• Exercise 3
• Exercise 3, jupyter notebooks

Suggested readings

• Gardiner W., Handbook of Stochastic Methods 1994, chapter 3
• Pawula R.,1967

Lecture 2: Overview of probability theory and statistics

• Lecture notes 2
• Exercise 2
• Exercise 2, jupyter notebooks

Suggested readings

• Gardiner W., Handbook of Stochastic Methods 1994, chapter 2

Lecture 1: Brownian motion, Einstein's theory, Langevin's theory

• Introduction to the course
• Lecture notes 1
• Exercise 1
• Exercise 1, jupyter notebooks

Suggested readings

• Gardiner W., Handbook of Stochastic Methods 1994, chapter 1
• Brown R. 1828
• Einstein A. 1905
• Langevin P. 1908
• Simulation of Brownian motion (video)

Figure. (a) Brownian motion; (b) Solution of the SIR model generated by Gillespie algorithm; (c) Solution of the Fokker-Planck equation generated by SqRA.