The goal of this course is to give an introduction to stochastic methods for the analysis and the study of complex physical, chemical, and biological systems, and their mathematical foundations.

- Random Variables, Distributions, and Densities
- Expected Values and Moments
- The Law of Large Numbers
- The Central Limit Theorem
- Conditional Probability and Conditional Expectation
- Monte Carlo Methods: Sampling and Monte Carlo integration
- Estimators, Estimates, and Sampling Distributions

- [1] A. Chorin and O. Hald, “Stochastic Tools for Mathematics and Science”, 3rd edition, Springer, 2013
- [2] L. Koralov and Ya. Sinai, Theory of probability and stochastic processes, 2nd edition, Springer, 2007

- Pseudorandom numbers
- Sampling random variables with given distribution
- Monte Carlo integration
- Estimators and estimates

- Discrete time Markov Chains
- Continuous time Markov Chains
- Representation of Energy Landscapes
- Markov Chain Monte Carlo Algorithms (Metropolis and Metropolis-Hastings)
- Transition Path Theory and Path Sampling Techniques
- Metastability and Spectral Theory

- [1] J. R. Norris, "Markov Chains", Cambridge University Press, 1998
- [2] Metzner, P., Schuette, Ch., Vanden-Eijnden, E.: Transition path theory for Markov jump processes. SIAM Multiscale Model. Simul. 7, 1192 – 1219 (2009)
- [3] A. Bovier, Metastability, in “Methods of Contemporary Statistical Mechanics”, (ed. R. Kotecky), LNM 1970, Springer, 2009
- [4] A. Chorin and O. Hald, “Stochastic Tools for Mathematics and Science”, 3rd edition, Springer, 2013

- Definition of Brownian Motion
- Brownian Motion and Heat Equation
- An Introduction to Stochastic Differential Equations (SDEs)
- Numberical integration of Stochastic ODEs: Euler-Maruyama, Milsteain's, MALA

- [1] A. Chorin and O. Hald, “Stochastic Tools for Mathematics and Science”, 3rd edition, Springer, 2013
- [2] Zeev Schuss, Theory and Applications of Stochastic Processes, An analytical approach, Springer, 2010
- [3] Grigorios Pavliotis, Stochastic processes and Applications, Diffusion Processes, the Fokker-Planck, and Langevin Equations, Springer, 2014
- [4] Desmond J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, 43, 3, (2001) 525-546

- The Freidlin-Wentzell Action Functional
- The Minimum Action Paths and the Minimum Energy Paths
- Methods for computing Minimum Energy Paths and saddle points

- [1] Freidlin, M. I. and Wentzell, A. D., Random Perturbations of Dynamical Systems, 2nd edition, Springer, New York, 1998, 3rd Edition, Springer, New York, 2013

- Diffusion maps
- Approximating differential operators by means of diffusion maps

- Principal component analysis (PCA)
- Multidimensional scaling (MDS)
- Diffusion maps
- Multiscale geometric methods
- Basics of Data Assimilation

- [1] K. Law, A. Stuart, K. Zygalakis, Data Assimilation: a mathematical introduction, Springer, 2015