Cornell Math - Math 672 (SP01)

Math 672 — Spring 2001
Stochastic Processes

1. Theory of stochastic interaction (conditional independence,Gibbs formula,Markov fields on graphs, Ising model-thermodynamic limit, Gaussian fields)
    
2. 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.
    
3. 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).
    
4. 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.