Probability and Cohomology

Research project P 14379 Probability and Cohomology Klaus SCHMIDT 08.05.2000 The purpose of this project is to use cohomological methods from ergodic theory to study certain problems in the theory of stochastic processes and their associated random walks, such as recurrence, exchangeability and characterization of tail-fields. Previous applications of cohomological ideas in this area have led to new recurrence conditions for random walks arising from stationary stochastic processes without any hypotheses about independence or existence of moments, or to exchangeability results (related to one of the two directions of de Finetti`s theorem) of considerable generality. These exchangeability results are examples of statements about tail-fields of two-sided stationary processes with values in locally compact abelian groups. Their extension to nonabelian groups has only been achieved under some very special assumptions (such as locally compact groups with finite or compact conjugacy classes). The extension to nilpotent groups is now known to be possible, but the next step (polycyclic groups) is still mysterious and will be one of the aims of the project. Another direction of this project will be the study of one-sided group-valued processes and their tail-fields, which is much more difficult than the two-sided case. Applications include ergodicity of horocycle flows and foliations in the infinite measure preserving case, as well as certain recurrence and ergodicity problems associated with random walks. Finally, the project hopes to achieve some insight into the nonstationary setting (e.g. reinforced random walks), but progress there is very difficult to predict.

The purpose of this project is to use cohomological and orbit equivalence methods from ergodic theory to study certain problems in the theory of stochastic processes and their associated random walks, such as recurrence, exchangeability and characterization of tail-fields. Cohomological methods have the advantage that they require no assumptions about integrability or existence of moments. To illustrate this point I should mention the following result from a joint paper by Gernot Greschonig and me: let X(n), n=1, be a real-valued stationary ergodic stochastic process, and let Y(n)=X(1)+...+X(n), n=1, be the associated random walk. If the distributions of the sequence Y(n)/n, n=1, do not diverge to infinity, then there exists a real constant c such that the random walk Y(n)-cn, n=1, is recurrent (i.e. returns to its starting position with probability 1). This result solves a problem which has been open for over 20 years and has since been published in the journal "Probability Theory and Related Fields". Two further preprints originating from this project solve an isomorphism problem in multi-parameter ergodic theory (i.e. in the theory of spatially extended systems with multi-dimensional symmetry groups). In a series of fundamental papers it was recently shown that, for certain systems, very weak forms of isomorphism imply surprisingly strong isomorphism properties (this phenomenon is called "isomorphism rigidity"). Here Siddhartha Bhattacharya and I could show that isomorphism rigidity holds for a much wider class of systems not amenable to traditional methods. These papers will appear in the "Israel Journal of Mathematics".