Rigorous topological dynamics for Gaussian processes and Brownian motion
Location: Hill 705
Date & time: Tuesday, 31 January 2023 at 11:00AM - 12:00PM
We consider the problem of understanding a dynamical system via finite samples of that system. The most straightforward approach to addressing this problem is to use the data to generate a model and then analyze the dynamics of that model using standard techniques. However, because long term dynamics may qualitatively change under arbitrarily small perturbations of a system it is unclear how reliable the conclusions that we arrive at in this manner will be; that is, it is difficult to quantify the probability that the predictions we make are correct.
In recent work, Batko et al. address this problem by combining Gaussian processes with multivalued dynamics. This talk will discuss this general framework as well as considering the special case of a Weiner process that is conditioned to pass through a finite set of points and the dynamics generated by iterating a sample path from this process. In both the general and special cases, topological techniques (Conley theory) are used to characterize the global dynamics and deduce the existence, structure and approximate location of invariant sets. Most importantly, these techniques determine the probability (or confidence) that this characterization is correct.