Analysis of complex systems related to fractional random fields

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This proposal will use novel mathematical techniques for analyzing the behavior of complex systems which evolve randomly in space and time, whose underlying structure is determined by a random field, and in particular, a random fractal. Random fields are useful tools for modeling spatial-temporal phenomena in physics, engineering, environmental sciences, and medical imaging. The proposal will contain projects related to the following two areas: (I) stochastic analysis; (II) limit theorems. (I) Stochastic analysis has been developed from Ito's seminal work, which introduced the stochastic integral with respect to the Brownian motion. This area includes the study of stochastic partial differential equations (SPDE's), whose goal is to offer a rigorous mathematical explanation for the behaviour of random phenomena which evolve over time. This program will analyze SPDE's perturbed by a Gaussian noise, which is more general than the space-time white noise, and in particular has the structure of the fractional Brownian motion in time (or in space). This study is motivated by the recent interest in random fractals, which are encountered frequently in a variety of applications. (II) The behaviour of the partial sum (or maximum) of independent observations lies at the origin of many investigations in probability theory, and is best understood via the invariance principles and strong approximations. In practice (for instance, in the case of financial data or geo-spatial data), most observations exhibit a certain dependence structure over time or space. The presence of a dependence structure between the observations usually alters significantly the validity of these results. This proposal aims at deriving new strong approximation results and invariance principles in the case of dependent observations arising from complex systems. In particular, linear sequences (also called moving average sequences) with stationary innovations constitute a large class of stationary sequences, which exhibit a special dependence structure and are suitable for modeling financial data or geo-spatial data.