Vovchanskyi M. Coalescing stochastic flows and point processes

The thesis is devoted to one-dimensional stochastic flows with coalescence, their approximations and related point processes. An analog of the iterated logarithm law for the size of a cluster is established for the Arratia flow at zero. The processes obtained by application of the fractional step method to the Brownian web are shown to be weakly convergent to an n-point motion of the Arratia flow with drift; an estimate of the speed of convergence is obtained in terms of the Wasserstein distance in the space of the distributions of random measures. Approximations of one class of coalescing Harris flows with solutions of stochastic differential equations are constructed; the convergence of the direct and dual flows and the convergence of images of the Lebesgue measure under the actions of such flows are established. Points densities that correspond to finite collections of starting points and specific realizations of the sequence of collisions are introduced for the Arratia flows. Representations of the point densities in terms of the Green functions of parabolic problems, Gaussian densities, Brownian bridges, stochastic exponentials for the Arratia flow and the distributions of vectors of the particles that have survived are given.