Notes from Winter 2020 can be found here.

Weeks 1-2: Poisson point processes

1. Definition; Poisson limits; additivity; motivating examples.
2. Thinning and labeling; conditional uniformity; characteristic functions and moments.
3. Statistics of spatial point processes: testing for independence.
4. Poisson jump processes.
5. The Cauchy process.
6. The Kolmogorov equation for pure-jump Levy processes.

Weeks 3-4: Markov chains and generators

1. Finite-state Markov chains: definitions, classificiation, transition probabilities.
2. Methods for simulation: Poissonization.
3. The approach to stationarity.
4. Generators for Markov chains and Poisson jump processes.
5. Uses and examples of generators.
6. Simple random walk and relatives.

Week 5: Brownian motion and Gaussian processes

1. Brownian motion as a limit of simple random walk via convergence of the graph Laplacian.
2. Brownian motion as an integral of white noise.
3. Other integrals of white noise and the L2 isometry.

Week 6: The stochastic integral

1. Itô integration.
2. Martingales and Itô integrals
3. Itô’s lemma and stochastic calculus

Weeks 7-8: Diffusions and associated PDE, via the Fokker-Planck equation

1. Stochastic differential equations
2. Backward Fokker-Planck equation from Itô’s lemma; connect to generators.
3. Forward Fokker-Planck equation.
4. Equilibrium distributions and approach to equilibrium.
5. First passage times of diffusions.
6. Detailed analysis of double-well potential example.

Weeks 9-10: Linear systems theory

1. Driven linear time-invariant systems LTI
2. State-space vs. input/output views of LTI systems.
3. Observability and controllability.
4. Eigen-spectra, poles and zeros.
5. Noise driven LTI systems (multi-dimensional Örnstein-Ulenbeck processes), steady-state autocorrelation and power spectral density.
6. Application to nonlinear dynamical systems close to bifurcations.