Kulyk O. Stochastic processes with smooth distributions

The thesis is devoted to the applications of the methods of infinite-dimensional stochastic analysis to investigation of the properties of the functionals on probability spaces of various kind. The change of variables formula for smooth measures on infinite-dimensional linear space is proved, and the integral representation for the functionals, analogous to Ito-Clark representation, is obtained. Analogues of exponential Khinchin inequalities and Fernique theorem are proved. The exact estimates of large deviations rate for smooth measures and distributions of their vector logarithmic derivatives are given in the terms of direction-wise averaging of the curvature of initial measure. The differential structure on the space of trajectories of a Levy process with arbitrary Levy measure, based on admissible "time-stretching" transformations, is given. This structure is used in the investigation of the local properties of distributions of the solutions of SDE's with jumps. The construction of a formal differentiation, correspondent to a given Markov process, is proposed. This approach is based on a "variation of generator method together with subsequent construction of the obtained parametric family of Markov processes on a same probability space. Applications of this construction to the problem of local hypoellipticity of singular diffusions are given.