Stochastic processes



  1. Basic Concepts . Brownian motion. Einstein's description. Langevin's description.
  2. Markov processes. The Chapman-Kolmogorov equation. Random walk. Stable distributions. Lévy flights.
  3. Stochastic differential equations. Wiener process. Continuous limit. Ito and Stratonovich interpretations. Orstein-Uhlenbeck process.
  4. Fokker-Planck equations. Derivation from the stochastic differential equation. Novikov's theorem. Stationary solution. Potential case. Detailed balance.
  5. Introduction to master equations. Two-state systems. Birth and death processes. Particle and occupancy number points of view. The step operator. The general form of a master equation. Examples.
  6. The generating function method for solving master equations The steady state. Large deviation theory.
  7. Approximate methods of solving the master equation. Gaussian approximation. Relationship with the Fokker-Planck equation.
  8. Van Kampen's expansion of the master equation. The singular expansion in an extensive parameter.