UC Davis

The mathematics which underly the intrinsic structures of stochastic processes and dynamics of probability are further developed, and broad applications are considered. I provide a general and rigorous definition of predictive states in stochastic processes, and demonstrate how they may be reliably and convergently estimated from time-series data. I connect this to new developments in the machine learning of dynamical systems. I further demonstrate that the dynamics of predictive states for a given stochastic process generates an algebraic structure, the observable semigroup, and show that this constrains the structure of physical systems which can generate said process. I apply this result to studying quantum machines which generate stochastic processes. By combining the algebra of the semigroup with that of majorization theory, I show that the constraints of the semigroup induce minimal costs in memory and energy required for these machines, and I compare these costs with classical machines, finding overall quantum advantage in memory but more ambiguous results in energy. I close by returning to questions of data science, and show how the mathematics of stochastic processes and majorization can help separate genuine structure from artifact in models of carbon footprints derived from global trade data.