TOPICS IN PROBABILITY: MATH 289B (WINTER 1997)
MEASURE VALUED DIFFUSIONS: FLEMING-VIOT PROCESSES
Professor: T. G. Kurtz (coinstructor - Professor R. J. Williams)
Lecture time: Thursdays, 2-3.50 p.m. (an extra hour for discussion will be arranged by Professor R. J. Williams for enrolled students). The room is to be arranged.

OUTLINE OF TOPICS TO BE COVERED

1. Generators and martingale problems for Markov processes.
Characterization of Markov processes as solutions of martingale problems. Proof of existence using weak compactness arguments.

2. Moran models and their measure-valued limits.
The Moran models of population genetics. Measure-valued Fleming-Viot processes obtained as limits of the finite particle models. Convergence arguments based on generators and martingale characterizations of the processes.

3. Duality for martingale problems.
Duality identity. Proof of uniqueness for martingale problems using duality. A dual for Fleming-Viot processes. Proof of ergodicity using duality.

4. Exchangeable processes and countably-infinite representations of measure-valued processes.
Ordered version of the Moran model. Equivalence of empirical measures of the two versions. Convergence of the ordered model to an infinite particle model. Representation of the Fleming-Viot process in terms of the infinite particle model. Applications of the infinite particle model to derive properties of the measure-valued process.

5. Particle representations for measure-valued processes.
Particle representations for more general measure-valued processes. Systems of SDEs for measure-valued processes with interaction.

6. Fleming-Viot processes with selection.
An infinite-particle model corresponding to the Fleming-Viot model with selection. Genealogy for models with selection.

7. Ergodicity and stationary distributions. Ergodicity for Fleming-Viot processes. Lyapunov and coupling methods. Conditions for absolute continuity of stationary distributions.

References (on reserve in S&E Library):

  • Ethier, S. N., and Kurtz, T. G., Markov Processes: Characterization and Convergence, John Wiley and Sons, 1986.
  • Dawson, D., Measure valued Markov processes, Ecole d'Ete de Probabilites de Saint-Flour XX, Lecture Notes in Math., 1541, Springer, New York, pp. 1--260.

    References (to be made available during the progress of the class):

  • Ethier S. N., and Kurtz, T. G., Fleming-Viot processes in population genetics, SIAM J. Control and Optimization, Vol. 31 (1993), 345-386.
  • Papers by P. Donnelly and T. G. Kurtz.

    Prerequisites:
    Math 280ABC or equivalent, or consent of instructor. Please direct any questions to Professor Ruth J. Williams, email: williams@math.ucsd.edu, phone: 534-6446.