Introduction to Random Dynamical Systems (AERO 630)
by Dr. Suman Chakravorty
Basics of probability theory
– Probability space and random variables
– CDF and PDF, transformation rule, joint, marginal and conditional distributions and densities
– Expectation and its properties, independence, conditional expectation, partition theorem, moments of a random vector: raw and central moments, correlation
– Some inequalities: CauchySchwarz, Holder, Jensen, Markov, Chebyshev
– Moment generating function and characteristic function, Central Limit Theorem
Discretetime stochastic process
– Random sequences, i.i.d. and i.i.p. process and examples (Bernoulli process and 1D random walk)
– Stationary, strictly stationary and widesense stationary (WSS) process
– I/O description of linear system: causality, BIBO stability, DFT and convolution, response of discrete LTI system subject to WSS input, PSD and autocorrelation
– White random sequence, ARMA model
– Conergence of a random sequence: sure, almost sure, in probability, meansquare sense, in distribution; Law of large numbers: WLLN and SLLN
– Markov chain on discrete statespace: transition kernel and probability, stationary distribution, homogeneous and inhomogeneous Markov chains, recurrent and transient states, positive and null recurrent states, aperiodic state, PerronFrobenius Theorem and irreducibility condition, ergodicity, Doeblin decomposition,
regionofattraction (ROA), celltocell mapping
Continuoustime stochastic process
– Review of preliminary concepts: probability space, algebra, probability measure, Borel algebra, radom variable and its PDF, stochastic process
– Dynamics of continoustime random variables, 1D Brownian motion, ItoSDE, an example: Langevin equation for Duffing oscillator
– Integration of SDEs, Ito version, Stratonovich version, strong and weak solution of SDEs
– Different approaches for weak solution: statistical/equivalent linearization, moment closure, equivalent nonlinear equation, perturbation methods (e.g. WKB expansion) for low intensity noise, Markov methods
– Markov methods: derivation of ChapmanKolmogorov equation, KramerMoyal expansion, derivation of FokkerPlanck equation (Kolmogorov's forward equation) and its properties
– Solving FokkerPlanck equation: stationary soluion: linear dynamics and Hamiltonianlike cases, rateofconvergence to stationary solution, transient solution: numerical approach and curseofdimensionality
