Mathematical Sciences: Structure of Stationary Stable and Other Infinitely Divisible Processes

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This project is intended to establish and explore new connections between the theory of certain infinite variance stationary processes and the ergodic theory of nonsingular flows. This connection leads to a natural classification of such random processes and gives a base for their systematic investigation. A new approach has been necessary because the ideas and techniques based on harmonic analysis of square integrable processes have a very limited use in the infinite variance case. On the other hand, many results of the ergodic theory of nonsingular flows have their natural counterparts in the theory of infinite variance stationary processes. For example, the famous Hopf decomposition of flows into dissipative and conservative parts permits us to identify and isolate the moving average parts of the corresponding stationary processes. Building upon these connections, structural and path properties of certain important classes of infinite variance stationary processes will be investigated. Stochastic stationarity can be observed in the long-time behavior of many random phenomena when the same patterns of behavior occur with time independent frequencies. For example, in a noisy communication channel the initial signal is superimposed with a random noise giving a stationary random process on the channel's output. In recent years there has been a growing interest in the investigating of highly random (infinite variance) models. The existing methodology and tools of harmonic analysis, that have been very successful in the study of moderately random stationary processes, have shown a very limited applicability in the infinite variance case. This project is intended to establish and explore new connections between certain classes of infinite variance stationary processes and the ergodic theory of nonsingular flows. This new approach reveals the structure of highly random stationary processes and it is anticipated that it may lead to a new methodology i n the study of such processes.