Korenovska Y. Geometric properties of infinite-dimensional maps generated by singular stochastic flows

The thesis is devoted to the study of random operators constructed by one-dimensional stochastic flows. Random operator $T_t$ of shift along an Arratia flow is a strong random operator in sense of A. V. Skorokhod. Consequently, images of compact sets under $T_t$ may not exist. It was obtained in this thesis that image of compact set under Gaussian strong random operator exists when Dudley condition holds. To obtain the sufficient condition in the case of $T_t$ it was proved in this work the formula of change of variables for an Arratia flow. Besides, the formula was used to show that compact sets may disappear under shift operator along an Arratia flow. It was constructed in this thesis a compact set in $L_2(\mathbb{R}),$ which almost surely doesn't disappear under $T_t.$ Kolmogorov $n$-widths of this compact were investigated, and it was shown how their changes under $T_t$ relate to properties of random integral operators generated by a point processes. It was shown in this work that correspond-\break ing random integral operators may be approximate by random nuclear operators. In the thesis we find which properties of clusters in Arratia flow influe on a rate of the approximation in the case of point process which is given by positions of particles in Arratia flow at a nonzero time.