Random Processes and Fields: Discrete Approximations, Special Wavelet-Based Decompositions and Simulation

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The principal investigator studies special wavelet decompositions ofrandom processes and fields. These representations have a multi-resolutionstructure and share other nice properties of standard waveletdecompositions. In contrast to the standard case, however, the highfrequency (small scale) terms in special wavelet decompositions have anelementary stochastic structure. Random processes and fields areconsidered under a wide range of assumptions: Gaussian, Gaussiansubordinated, stable, stationary, self-similar, some non-stationary, andothers. In addition to establishing and understanding special waveletdecompositions, questions regarding their use in simulation, analysis andmodeling are also addressed.The project adds to the recent growth of the multi-scale(multi-resolution) approach where observed phenomena are considered atdifferent scales (resolutions). This approach where wavelets play acentral role, has found successful applications in numerous fields such asmedical imaging, turbulence, de-noising, astronomical and Internet trafficdata analysis. The goal of the project is to develop a newmulti-resolution framework that is suitably adapted to random phenomena.Despite some progress, such multi-resolution framework has been missing ata fundamental level. The developed framework should provide novel analysistools and conceptual perspectives, for example, establish new connectionsbetween models in discrete and continuous times, shed light on simulation,which has become an indispensable tool of any scientific research, or beuseful in modeling through the multi-scale approach. The project shouldalso be of direct interest to and enhance ties between researchers in manyareas of Science such as Applied and Pure Mathematics, Physics, SignalProcessing, Probability and Statistics.