Fomichov V. Evolution of diffeomorphic Brownian stochastic flows

In the thesis we study the properties of Brownian stochastic flows, which serve as a mathematical model in describing the motion of linearly ordered systems of interacting particles. We found the asymptotics of the moments of the distance between particles in Harris flows and the asymptotics of the moments of their n-point motions. We established the weak convergence of the n-point motions of Harris flows as the interaction between particles becomes singular to the n-point motions of the Arratia flow. We obtained an estimate for the Wasserstein distance between the distributions of a measure transported by a Harris flow with near-singular interaction between particles and the Arratia flow. We computed the level-crossing intensity of the density of the image of the Lebesgue measure under the action of a diffeomorphic Brownian stochastic flow and established its asymptotic behaviour as the height of the level tends to infinity. We found the distribution of the number of clusters in the Arratia flow.