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 basic theory underlying stochastic processes and advanced numerical methods to solve problems with real applications.
Program:
1. Introduction to Brownian motion and random walk. Derivation of the Fick’s laws for diffusive processes.
2. The Wiener process, introduction to stochastic differential equations, the Ito and Stratonovich integral, the Ito’s formula.
3. Markov processes, the Chapman-Kolmogorov equation and the master equation.
4. The Langevin equation and the fluctuation–dissipation theorem.
5. Derivation of the Fokker-Planck equation from the Langevin equation.
6. The Kramers-Moyal expansion and the relationship between the master equation and the Fokker-Planck equation.
7. The Ornstein Uhlenbeck Process.
8. The Kramers’escape problem.
9. The Poisson process and the birth-death process.
Methods:
• The Euler-Maruyama and the Euler-Heun schemes to solve stochastic differential equations respectively
for Ito and Stratonivich interpretation.
• The Square Root Approximation of the Fokker-Planck operator.
• Markov-state models (MSMs).
• Robust Perron Cluster Analysis (PCCA+).
• The Gillespie algorithm to solve a system of stochastic equations provided the reaction rates.
Applications:
• Low dimensional metastable systems.
• The Black-Scholes model.
• Epidemiological models such as the SIR model.
• Analysis of real Molecular Dynamics trajectories.