Stochastic Processes with Applications
A broad introduction to stochastic processes, focusing on their application to describe natural phenomena and on numerical simulations rather than on mathematical formalism. Define and classify stochastic processes (discrete/continuous time and space, Markov property, and forward and backward dynamics). Explore common stochastic processes (Markov chains, Master equations, Langevin equations) and their key applications in physics, biology, and neuroscience. Use mathematical techniques to analyze stochastic processes and simulate discrete and continuous stochastic processes using Python.
The course is aimed at students interested in modeling systems characterized by stochastic dynamics in different branches of science.
- Basic concepts of probability theory. Discrete and continuous distributions, main properties. Moments and generating functions. Random number generators.
- Definition of a stochastic process and classification of stochastic processes. Markov chains. Concept of ergodicity. Branching processes and Wright-Fisher model in population genetics.
- Master equations, main properties and techniques of solution. Gillespie algorithm. Stochastic chemical kinetics.
- Fokker-Planck equations and Langevin equations. Main methods of solution. Simulation of Langevin equations. Colloidal particles in physics.
- First passage-time problems. Concept of absorbing state and main methods of solution. First passage times in integrate-and-fire neurons.
- Element of stochastic thermodynamics. Work, heat, and entropy production of a stochastic trajectory. Fluctuation relations, Crooks and Jarzynski relations.
Reports (numerical simulations): 60% hands-on sessions, 20% homework assignments, 20% participation in class
- “Random walks in Biology” by H. C. Berg (1993) Princeton University Press
- “Stochastic Methods: A Handbook for the Natural and Social Sciences” by C. Gardiner (2009) Springer
- “An Introduction to Probability Theory and its Applications, Vol 1” by W. Feller (1968) Wiley
- “The Fokker-Planck Equation”, by H. Risken (1984) Springer
Calculus, Fourier transforms, probability theory, scientific programming in Python.
Students must install the Jupiter notebook system