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Probability and statistics: Joint and conditional probabilities and densities. Moments, cumulants, generating functions, characteristic function. Binomial, Poisson, Gaussian distributions. Stable distributions, limit theorems, diffusion limit of random flights.  Infinitely divisible distributions.

Stochastic processes: Discrete and continuous random processes. Joint and conditional probability distributions. Autocorrelation function. Markov chains. Discrete Markov processes, master equation. Poisson process,  birth-and-death processes. Jump processes. Correlation functions,  power spectra.  Campbell's Theorem,  Carson's Theorem.  Thermal, shot, Barkhausen and  1/f  noise.

Continuous Markov processes:  Chapman-Kolmogorov equation, transition rate, Kramers-Moyal expansion. Fokker-Planck equation, backward Kolmogorov equation, first passage   and exit time problems.  Level-crossing statistics.

Stochastic differential equations:  Langevin equation, diffusion processes, Brownian motion, role of dimensionality, fractal properties.

Random walks: Markovian random walks.  Random walks and electrical networks,  random walks in biology. Levy flights. Self-avoiding walks and polymer dynamics.  Random walks on fractals.  Non-Markov continuous time  random walks. 

Randomness in deterministic dynamics:  Coarse-grained dynamics, Markov and generating partitions,  recurrence statistics.

 

 

Elementary notions of probability and statistics


  • Balakrishnan V: Elements of Nonequilibrium Statistical Mechanics (Ane Books). Beck C and Schogl F: Thermodynamics of Chaotic Systems (Cambridge University Press).
  • Berg H C: Random Walks in Biology (Princeton University Press).
  • Cox D R and Miller H D: The Theory of Stochastic Processes (Chapman and Hall).
  • Denker M and Woyczynski W A: Introductory Statistics and Random Phenomena (Birkhauser).
  • Doi M and Edwards S F: The Theory of Polymer Dynamics (Cambridge University Press).
  • Doyle P G and Snell J L: Random Walks and Electrical Networks (Mathematical Association of America).
  • Gardiner C W: Handbook of Stochastic Processes (Springer).
  • Grimmett G and Stirzaker D: Probability and Random Processes (Oxford University Press).
  • Kac M: Probability and Related Topics in Physical Sciences (Wiley-Interscience).
  • Papoulis A: Probability, Random Variables and Stochastic Processes (McGraw- Hill).
  • Risken H: The Fokker-Planck Equation: Methods of Solution and Applications (Springer).
  • Stratonovich R L : Topics in the Theory of Random Noise, Vols. 1 and 2 (Gordon and Breach).
  • Van Kampen N G: Stochastic Processes in Physics and Chemistry (North-Holland).
  • Wax N: Selected Papers in Noise and Stochastic Processes (Dover).
  • Weiss G H: Aspects and Applications of the Random Walk (North-Holland).
  • Wong E: Introduction to Random Processes (Springer).


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