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
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