Cornell Math - MATH 778, Spring 2000

MATH 778 — Spring 2000
Markov Processes

1. Brownian motion. Additive functionals and transformations based on them (killing, random time change, Girsanov's transformation). Local times and self-intersection local times. A general continuous strong Markov process in R.
   
2. Three methods of constructing a diffusion: via fundamental solution of the heat equation, by Ito's equation, by solving a martingale problem.
   
3. The first boundary value problem for linear elliptic and parabolic equation in an arbitrary domain. Analytic solution by the Perron-Wiener-Brelot method. Probabilistic solution. Martin boundary. Stochastic boundary values.
   
4. Exceptional sets in analysis and in probability. Probabilistic theory of potential.
   
5. Infinitely divisible random measures.
   
6. Branching exit Markov systems.
   
7. Superdiffusions and semilinear elliptic and parabolic equations. Removable singularities in a domain and on its boundary — relation to hitting probabilities by a superdiffusion. Characterization of positive solutions by their trace on the boundary.

I will avoid generalities and technicalities and concentrate on principal ideas and concrete problems. Prerequisites: the Lebesgue integration theory and an interest in probability theory.