Non-Stationary Random Dynamical Systems and Applications

Random and unpredictable factors are always present and cannot be completely avoided. Therefore, the development of numerous real-life systems can be more accurately explained through the application of distinct maps, that is, of transformations that vary with time, rather than by repeatedly applying the exact same transformation. It is intuitive to examine these issues using typical sequences of maps, which brings us to the concept of random dynamics. Theory of random dynamical systems goes back to Ulam and von Neumann, as well as to Kakutani. Since then, it turned into a well-developed area of mathematics that has numerous applications. But what if the distribution and strength of the random factors itself does not remain constant, but changes with time? That can be modeled by non-stationary random dynamical systems, and the study of the properties of such systems and of their applications is the main goal of this project. Mentoring students will be an essential part of the project. Numerous questions closely related to the proposed project will be suggested to graduate and undergraduate students initiating and increasing their involvement in research in mathematics. In the first part of the project the goal will be to show that a smooth random dynamical system, under an assumption of absence of invariant measures, must lead to almost sure exponential growth of the norms of random compositions, even in non-stationary cases. Also, moduli of continuity of stationary measures will be studied. The second part of the project will lead to a non-stationary version of the Furstenberg Theorem on random matrix products, that claims that there exists a nonrandom sequence, a non-stationary analog of the Lyapunov exponent, that almost surely describes the behavior of the norms of random matrix products. Non-commutative versions of the Central Limit Theorem, the Iterated Logarithm Law, and other limit theorems will be proven in non-stationary settings, for random matrix products, and, more generally, for random walks on Lie groups. The third part of the project will provide the proof of Spectral and Dynamical Localization for the non-stationary Anderson Model (including the Anderson-Bernoulli case). Besides, the questions on topological structure of the spectrum and essential spectrum of the non-stationary Anderson Model will be addressed. In particular, it will be shown that for a discrete Schrödinger operator with potential given by a sequence of independent identically distributed random variables plus a quasiperiodic background the spectrum must consist of a finite number of intervals, while for a periodic background it can have infinitely many gaps. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.