Geometrical characteristics of curves in a random medium

The talk is about the model of evolution of curves in random medium based on stochastic interaction equations introduced by A.A. Dorogovtsev (Stochastic interaction flows and measure-valued Markov processes. Doklady RAS,388(2003),N2, pp.151-154). As the geometrical characteristics the local self-intersection times are considered. For the definition of these local times for an image of the Brownian trajectory under the random map the generalization of Dynkin renormalization is needed (Dorogovtsev A., Izyumtseva O., Hilbert-valued self-intersection local times for planar Brownian motion. ArXiv: 1708.00608). The sufficient conditions for imbedding of compact into a Hilbert cube due to A.A. Dorogovtsev ( Dorogovtsev A., Popov M., Geometric entropy in Banach spaces. Theory of Stochastic Processes, 35(2014),N2, pp. 10-30) are used. The last part of the talk contains the connection between the mutual rotation angle and local intersection time of two correlated Brownian motions.