Karen Gunderson

(Heilbronn Institute for Mathematical Research, University of Bristol)

Random Markov Processes

Date Friday, January 10, 2014

In 1990, Kalikow introduced the notion of ''random Markov processes'' to refine classifications of measure-preserving transformations on arbitrary measure spaces. Looking at the shift operator on probability spaces of infinite sequences of some set of states, a random Markov process is a stochastic process for which there is a coupling of the sequences of states with doubly infinite sequences (m(i)), so that for every i, the distribution on the states at the i-th step of the process depends only on the m(i) previous states. Random Markov processes are a generalization of usual Markov chains and have been used to produce counter-examples to questions about uniqueness of certain types of measure spaces and as extensions for non-periodic transformations.

In this talk, I will discuss some new results on the classification of random Markov processes on any finite number of states and extensions of these results to certain processes with infinitely many states. In addition, I will give some examples showing that further conditions are needed to extend results about finite numbers of states to those with even countably many states.

Joint work with Neal Bushaw (Arizona) and Steve Kalikow (Memphis).