Stochastic Evolution of Complex Shapes and Markov Dynamics in Configuration Spaces

This project is aimed at the study of the deterministic and stochastic evolution in the space of complex shapes. Our methodology is based on the approach in which the shape evolution is given by means of the Loewner-Kufarev equation. This gives an opportu nity of a constructive description of moving shapes generated in a canonical way by a given configuration of points in the complex plane, which can be stochastic itself, or even be given by a stochastic point process. The next goal of the project to study the influence of the environment and of the environmental potential on this evolution, and to understand the information on statistical properties of the environment encoded by limiting shapes. The development of this program assumes a special study of t he space of shapes from the analytic, differential geometric and probabilistic viewpoint. On the other hand this approach incorporates recent developments in the theory of configuration spaces and stochastic dynamics of configurations. Our approach will g ive us a way to construct measures on the space of shapes given measures on the space of configurations, on which they are known by methods of statistical physics and probability theory.