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Physics - Physical Applications of Stochastic Processes

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 pisible 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. Faculty Prof. V. Balakrishnan, IIT Madras Courtesy: NPTEL HRD

Course Objective

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 pisible 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. Faculty Prof. V. Balakrishnan, IIT Madras Courtesy: NPTEL HRD

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