Fractionality: entropy and related physical properties

This scientific project is devoted to the study of one of the fundamental physical and mathematical concepts, entropy, that is, a measure of the chaos of the surrounding world. Entropy is closely connected with the amount of information that the dynamic system under study carries. In turn, the entropy of a system that operates in a stationary mode depends on the presence, absence, and, if available, on the amount of memory that the system accumulates over time. The memory of a physical system is a comparatively modern phenomena observed in economy, finance, technical devices. Moreover, the relationship between entropy and memory of the system is not linear. One of the main tasks of the project is to establish the dependence of entropy on the presence and accumulation of memory in the system and to learn how to manage this dependence. Then it will be possible to apply the entropy control to physical and economical problems.

The project is devoted to advanced studies of fractional processes. These processes include: fractional Brownian motion, fractional Gaussian noise, tempered fractional Brownian motion, Gaussian-Volterra processes, multifractional Brownian motion, multiparameter fractional fields.

Project objective

The main efforts will be focused on four problems:

1. To study entropy of fractional processes as a function of their parameters, with physical applications.
2. To study the properties of Cholesky decomposition of covariance matrix of fractional Brownian noise, and projections of fractional noise onto its individual coordinates.
3. Parameter estimation for the advanced models involving fBm, especially with irregular drift coefficient.
4. Ergodic theorems for fractional diffusion processes, including rate of convergence.