The lecture course is intended for students of physics, especially in the first or third semester of their Master Study Course. It will provide an introduction into the basics of the theory of stochastic processes from the view of a physicist and the formal tools to understand a broad field of physical phenomena. The knowledge of concepts of ststatistical physics may be helpful but is not supposed. Topics include:

• Random variables and stochastic processes: Kolmogorov axioms, Limit laws, Large deviations, Classification

• Markov processes: Chapman-Kolmogorov equation, Master equation, Kramers-Moyal expansion, Diffusion processes, Fokker-Planck equation

• Continuous stochastic processes: Gaussian processes, Ornstein-Uhlenbeck process, White noise, Wiener process

• Lévy processes: Stable probability distributions, Fractal dimension of the Wiener-Lévy process

• Discrete stochastic processes: Poisson events, Dichotomous Markov process, Kubo-Anderson process, Kangaroo process

• Martingales

• Langevin equations and Fokker-Planck equations: Stochastic differential equations and stochastic integrals (Ito vs. Stratonovich), Stochastic Liouville equation, Exact theorems for averaging (Furutsu-Novikov, Shapiro-Loginov)

A number of applications will be discussed in due course. They include Brownian motion and (anomalous) diffusion, mean first passage times, noise induced phenomena (Kubo's theory of ESR linewidth, stochastic resonance, Brownian motors, non-equilibrium phase transitions, on-off intermittency), the Black-Scholes theory, and the numerical simulation of stochastic processes.

The credit points are given if an oral exam of about 45 min is passed.

Exam at 18 and 23 February 2015 pdf

Recommended literature pdf

Links of interest