Stochastic processes

Responsible: PD Benjamin Friedrich
Hours per week (2)
Language: English

§1 Introduction: Diffusion
- Random forces, mean-square displacements, Einstein relation
- Ergodic hypothesis
- Application areas of stochastic processes
- Programming exercise: Fluctuating pressure in an ideal gas

§2 Fundamental concepts
- Short review of probability theory, probabilities and probability densities, moments and cumulants
- Bayes formula
- Most important probability distributions (Binomial distribution, Poisson distribution, normal distribution, power-law distributions)
- Central limit theorem (with heuristic proof)
- Stochastic processes: time-discrete and time-continuous
- Markov processes and transition probabilities
- Gaussian white noise and Wiener processes
- Programming exercise: Visualization of Central limit theorem

§3 The Fokker-Planck equation
- Chapman-Kolmogorov equation
- Derivation: weak formulation using smooth test function, Taylor expansion of test function, Kramers-Moyal coefficients as moments of time evolution operator, integration by parts
- Example: moments for Wiener process
- Fokker-Planck equation for the Langevin equation with non-multiplicative noise
- Programming exercise: Equilibration of a diffusive particle in a bistable potential

§4 Diffusion to capture
- First passage times for Wiener process
- Diffusion to an absorbing target, electrostatic analogy
- Polya’s theorem
- Programming exercise: Recurrence of random walks

§5 Kramer’s escape rate theory
- Role of boundary conditions, first-passage times (FPT), differential equation for FPT
- Derivation of Kramer’s rates
- Spectral representation of the Fokker-Planck operator, eigenfunctions, decay times
- Noise-induced oscillations in excitable systems
- Programming exercise: Stochastic transitions over a potential barrier

§6 Synchronization of noisy oscillators
- Noisy oscillators, phase correlation function, half-width-at-half-maximum, quality factor
- Adler equation
- Diffusion in tilted potential, phase-slip rates (Stratonovich), giant diffusion
- Stochastic resonance: the periodically driven bistable potential Programming exercise: Kuramoto model with noise

§7 The fluctuation-dissipation theorem
- Examples: Nyquist noise
- Fluctuation spectra and linear response theory
- Wiener-Khinchine theorem
- Derivation
- Application: Calibrating optical tweezers

§8 Ito versus Stratonovich calculus
- Stochastic differential equations with multiplicative noise
- Explicit versus semi-implicit integration (Euler and Euler-Heun schemes)
- Noise-induced drift
Programming exercise: Thermophoresis (Soret effect)

§9 Rotational diffusion
- The 2D- and 3D-rotor
- Application: dielectric relaxation
Programming exercise: Anisotropic diffusion

§10 Noise in chemical reactions
- Gillespie algorithm
- Diffusion approximation
- Berg&Purcell: Noise in chemo-sensation

§11 Kalman filters
- Linear dynamic models, measurement models
- Optimal update rule

§12 Statistical testing and decision making
- Hypothesis testing, ROC, significance levels
- Maximum-Likelihood estimates, the rationale behind least-square fitting, Fisher information matrix, confidence intervals
- Bayesian parameter estimation

§13 Further topics
- Kinetic proof reading
- Chemotaxis along biased random walks
- Financial markets

Background needed:
- Multivariate calculus
- Elementary probability theory
- Ordinary and partial differential equations
- Basic programming skills

Target audience:
- Physics students at the Master’s level
- Mathematics students interested in applications of stochastic processes
- Bioengineering students with strong background in quantitative methods

Relevant literature:
[1] H. Risken: The Fokker-Planck Equation: Methods of Solution and Applications, Springer
[2] vanKampen: Stochastic Processes in Physics and Chemistry, North-Holland
[3] Philip Nelson: Physical Models of Living Systems, W. H. Freeman, 2015
[4] W. Paul und J. Baschnagel: Stochastic Processes: From Physics to Finance, Springer