Isotropic Stable Random Fields and Infinite Dimensional Stochastic Integrals

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0204992Rosinski A novel approach, based on infinite dimensional group representations, is proposed for the analysis of isotropic and isotropic homogeneous stable random fields. There are many analogies between problems considered in this project and representation theory in quantum physics, in particular, Mackey's theory. This project is intended to build on these analogies to establish spectral representation of isotropic stable random fields in possibly the most explicit form that could be used for analysis, modeling and computer simulation of such random fields. The second part of this proposal is concerned with infinite dimensional stochastic integrals. Various sample path properties of stochastic processes, such as continuity and boundedness, can often be described in terms of the existence of such infinite dimensional stochastic integrals. It is proposed to simplify and unify the present approach to infinite dimensional stochastic integrals of deterministic functions and then to extend these results to nonanticipating integrands using decoupling techniques, among others. This is an outgrowth of the present research of the PI with Professor Michael B. Marcus. Development of stochastic integrals with respect to cylindrical semimartingales in Hilbert spaces is also proposed. This development will follow a recent progress of the PI and his collaborators on the radonification of cylindrical semimartingales. Isotropic random fields play an important role in the statistical theory of turbulence and other areas of natural sciences and engineering. They can be used to model 3-dimensional turbulent velocity, velocity and pressure, etc. The classical theory of isotropic random fields is not suitable for modeling random phenomena with long range dependence and high variability, that are often observed. Therefore, the PI proposes to investigate isotropic random fields with heavy tailed stable distributions. Puzzling connections with certain methods of quantum physics are found under this approach; they will be further investigated and exploited. The second part of this proposal is concerned with the theory and applications of infinite dimensional stochastic integrals. The importance and need for the study of such integrals comes from the theory of random systems characterized by very large or continuous set of parameters. Infinite dimensional approach makes such systems simpler and mathematically tractable.