Rigorous computational methods for understanding the statistical behavior of random dynamical systems.

In 2019 it has been estimated that the amount of digital data in the world is 40 zettabytes, 40 times the number of observable stars in the universe. Obviously providing an accurate estimate of this amount is impossible, but looking at our society, it is not difficult to imagine that it is so. Managing this amount of data and digital devices is a huge challenge for mankind, involving virtually every area of knowledge, from metallurgy to ethics. It arises spontaneously then, to wonder if, despite their intrinsic complexity, it is possible to understand, in some sense, the evolution of this type of complex systems. This question is naturally placed in the mathematical field, and one of the techniques currently used to model an observable phenomenon is to describe the evolution of a system through the evolution of a certain stochastic process, a random dynamical system. The purpose of this thesis is to try to answer the following question: given a random dynamical system, are we able to predict (not just simulate) its behavior in the long term? If I put a drop of black ink in a glass of water, the short term interactions between these two liquids are unpredictable. However, we can all say with certainty that after enough time, the water and ink mix completely, the glass is filled with a grey liquid, the system will have reached its equilibrium state. The example just exposed, can be formalized in many ways, one of these is to consider a certain operator defined on the appropriate spaces, which maps measures to measures, and prove for example that this operator admits as fixed point the Lebesgue’s measure properly normalized (the state of equilibrium). This type of operators, are called Transfer Operators, and are linear operators defined on appropriate Banach spaces, and are the object of study of this thesis. A transfer operator associated with a dynamical system describes the evolution of the densities according to the dynamics (in the previous example, the drop of ink that we can model with a delta of Dirac, will evolve in the measure of Lebesgue). Under appropriate assumptions, these transfer operators admit fixed points, which are the stationary measures of the system, and our goal is to calculate these fixed points, which represent the statistic behavior of the system under consideration after a long time. Unfortunately, in many cases, even simple ones, the calculation of the stationary measure, or better, of its density with respect to the Lebesgue measure, cannot be addressed analytically. The strategy we follow is therefore to rigorously approximate these densities; we remark on the importance of the fact that we are interested in having rigorous quantitative estimates, and not in numerically simulating a system. The central core of this interdisciplinary thesis is to use mathematical ideas that allow us to have an explicit estimation of the errors related to these rigorous approximations, combined with an accurate computer science implementation, which leads to have quantitative theorems proved with the aid of a computer.