Kuznetsov V. Geometrical properties of stochastic flows

The thesis is devoted to the investigation of geometrical properties of random objects: random fields, stochastic flows, random braids. We obtain the expression for the first and the second index of the field in terms of the conditional density of two-point distributions. For an isotropic Gaussian random field the first and the second moments of the index are found in terms of the covariance function of the fields' components. We obtain the representation of the Vassiliev's invariants for braids formed by trajectories of two-dimensional continuous semimartingales with respect to the common filtration in the form of multiple Stratonovich integrals. We find the common asymptotical distribution of the mutual winding angles in Brownian flows with zero top Lyapunov exponent. We show that for the winding angle of the Brownian motion around the origin the weak large-deviation principle holds, but the full does not. Also, we show that the estimations of the large-deviation principle hold on the class of cylindrical sets in $C([0,1])$.