Cornell Math - Math 672 (SP01)
Math 672 — Spring 2001
- Theory of stochastic interaction (conditional independence,Gibbs formula,Markov fields on graphs, Ising model-thermodynamic limit, Gaussian fields)
- Markov chains in a general measurable state space, asymptotic behavior at large time ( ergodicity coefficient, weak and strong ergodicity), the case of a finite and countable state space, random walks. Stopping times, strong Markov property, Doeblin's method, renewal theorem.
- Brownian motion (passage to the limit from random walks, construction of continuous Brownian motion, invariance and self-similarity properties, first exit times and exit distributions, Markov and strong Markov properties, Blumenthal's 0-1 law,probabilistic solution of the Dirichletr problem).
- Martingales (the Doob-Meyer decomposition, Doob's upcrossing inequality, Kolmogorov's inequality, Hilbert space of continuous martingales). Stochastic integrals, stochastic differential equations, Ito's formula, diffusions.