The main focus of this research is on mathematical theory and methods which have a direct bearing on problems involving multiscale phenomena in multiplicative structures that arise naturally in a wide variety of modern applications, such as river basin hydrology, fluid turbulence, financial securities markets, spin glasses etc. The essential mathematical ingredients are a space-time random field defined by a multiplicative cascade random process and a random network represented by a random tree graph, together with a system of equations directing the evolution. Much of the data collected and reported on these structures is in the form of log-log plots of some quantity versus a length scale. This leads to the introduction and refinements of new classes of self-similar spatial/temporal models whose scaling structure is inferred from empirically observed sample realizations. Thus we seek to calculate certain large-sample (fine scale) limits of statistical estimators of exponents and limit laws governing fluctuations. In addition, self-similarities and scaling exponents are sought for the cascade and network models. Then connections between the scaling exponents of the flow processes and those of the cascade and network exponents may be investigated. The prospect of a theory which computes structure functions, (i.e., multiscaling exponents) for extreme flows from corresponding structure function calculations on the inputs and network defines the frontiers of this research.
A fundamental problem from environmental science is to determine the structure of river flows (e.g. extremes) from a basin given data on the local climate (e.g. rainfall) and topography (river network structure, soil moisture). In most parts of the world the information available for the planning of dams, flood insurance, military tactics etc. is in the form of remotely sensed local climate and topography. The mathematical formulation is based on a stochastic system of conservation equations (mass, momentum) which relate the flows to the multiplicative stochastic rainfall inputs and complex network topography via scaling and multiscaling exponents which are estimated from remotely sensed data. One of the practical aspects of results of this type is to assist hydrologists and engineers in extrapolating localized observations to larger scales, and to regionalize predicted flows. However, the broad mathematical framework contributes to our understanding of diverse natural stochastic phenomena such as fluid turbulence, stochastic investment yields, renewable natural resource distributions, spin glass magnets etc., which are intrinsically multiplicative in space and/or time.
