RESEARCH IN PAIRS
Martingale Techniques and Anisotropic Banach Spaces
August 7 - 18, 2017
The main objective is to explore the applicability of martingale techniques in the setting of invertible systems using the recently developed ‘adapted’ Banach spaces pioneered by Liverani, Baladi, Gouëzel, Tsujii, Keller, Demers and others.
The application of martingale theory to the study of statistical properties of dy- namical systems was pioneered by M. Gordin in 1969. By studying the action of the transfer operator (the dual to the action of composing an observable with the transformation) the Gordin approach expresses an observable on the system as the sum of a reverse martingale difference and a coboundary.
Recent work has investigated the action of the transfer operator directly on certain anisotropic Banach spaces. This approach avoid introducing symbolic coding, and allows perturbation of the system. However, the martingale technique of Gordin has not yet been adapted to this context although in certain situations it enables one to prove stronger results. Examples include convergence of stochastic integrals and rates of convergence in the weak invariance principle, where the primary existing probabilistic tools are via martingale theory. The collaboration will be to develop ways of modifying Gordin’s approach to the setting of anisotropic Banach spaces.
Mark Demers (Fairfield University)
Ian Melbourne (University of Warwick)
Matthew Nicol (University of Houston)