## 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

**Contact:**

benjamin.m.friedrich@tu-dresden.de

https://cfaed.tu-dresden.de/friedrich-group-teaching