Hlyniana K. Discrete-time stochastic flow

The thesis is devoted to the investigation of stochastic flows of Brownian motions with singular interactions and their analogue with discrete time. Stationarity and ergodicity with respect to the spatial variable for the discrete-time stochastic flows are established. Discrete-time stochastic flows that approximate the Arratia flow were considered. For such flows asymptotics of convergence to zero of the time when two particles disobey the initial order was obtained. An analogue of Krylov-Veretennikov expansion for the discrete-time flow is obtained. A core for the generator of the m-point motion semigroup for the Arratia flow is discovered. The action of the m-point motion semigroup is written in terms of sum of integrals where each summand is indexed by a binary forest.