
Lecture Description
- The CosmoLearning Team
Course Index
- Discrete Probability Distributions (Part 1)
- Discrete Probability Distributions (Part 2)
- Continuous Random Variables
- Central Limit Theorem
- Stable Distributions
- Stochastic Processes
- Markov Processes (Part 1)
- Markov Processes (Part 2)
- Markov Processes (Part 3)
- Birth-and-Death Processes
- Continuous Markov Processes
- Langevin Dynamics (Part 1)
- Langevin Dynamics (Part 2)
- Langevin Dynamics (Part 3)
- Langevin Dynamics (Part 4)
- Ito and Fokker-Planck Equations for Diffusion Processes
- Level-Crossing Statistics of a Continuous Random Process
- Diffusion of a Charged Particle in a Magnetic Field
- Power Spectrum of Noise
- Elements of Linear Response Theory
- Random Pulse Sequences
- Dichotomous Diffusion
- First Passage Time (Part 1)
- First Passage Time (Part 2)
- First Passage and Recurrence in Markov Chains
- Recurrent and Transient Random Walks
- Non-Markovian Random Walks
- Statistical Aspects of Deterministic Dynamics (Part 1)
- Statistical Aspects of Deterministic Dynamics (Part 2)
Course Description
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.