Hopkalo O. Conditions of existence and properties of solutions of boundary value problems with random initial conditions

The thesis is devoted to the investigation of sample paths properties of stochastic processes belonging to Orlicz spaces of random variables, as well as, processes from certain spaces of Orlicz processes of exponential type and processes related to partial differential equations with random initial conditions. The study is based on the use of the entropy method. The origins of this approach can be found in the paper by R. Dudley (1967), where sufficient conditions for the boundedness of Gaussian processes were based on the corresponding entropy integrals. Further these ideas were extended by X. Fernique (1975) and M. Ledoux and M. Talagrand (1991). The applications of entropy methods for more general classes of processes were presented in the monograph by V. Buldygin and Yu. Kozachenko (2000). The monograph investigates in detail the properties of trajectories of stochastic processes taking values in the Orlicz spaces of random variables. Considerable attention is paid to the subclass of processes from Orlich spaces of exponential type, which are called φ-sub-Gaussian processes and generalize Gaussian and sub-Gaussian ones. The study of the properties of trajectories of stochastic processes, distributions of functionals of stochastic processes, derivation of estimates for moments and distributions of suprema are important directions of development of the theory of stochastic processes, especially in connection with practical needs in various applied areas. One of the important applications of results on properties of trajectories of stochastic processes and results on the distributions of their suprema is in the study of solutions of partial differential equations with random initial conditions. The practical component in such problems is to establish the dependence of the behavior of solutions on the corresponding initial conditions. In the thesis the properties of sample paths of stochastic processes from Orlicz spaces defined on unbounded domains are investigated, the conditions of boundedness and continuity with probability 1 are established and estimates of distributions of suprema of such processes are obtained. Estimates of the distributions of suprema of stochastic processes related to the heat equation with random initial conditions from the Orlicz spaces are established. New estimates are obtained for the distributions of the supremum of increments of φ-sub-Gaussian stochastic processes defined on a compact set. The properties of the trajectories of φ-sub-Gaussian random processes defined on unbounded sets are investigated, the conditions of boundedness and continuity with probability 1 are established, and estimates for the supremum of distributions of such processes are obtained. Estimates for the distributions of suprema of stochastic processes associated with the heat equation with φ-sub-Gaussian random initial conditions are established. The rate of growth of trajectories of φ-sub-Gaussian processes is investigated. The properties of φ-sub-Gaussian processes related to Cauchy problems for the heat equation with φ-sub-Gaussian stationary random initial conditions are investigated. Modules of continuity of solutions for different classes of initial conditions, estimates for supremum of distributions and the rate of growth of normalized solutions are obtained. The results are generalized for a wider classes of φ-sub-Gaussian initial conditions. A generalized Levy-Baxter theorem for pseudo-Gaussian random vectors is established. Generalized estimates for the distribution of quadratic form of these vectors are obtained. The thesis has both theoretical and practical significance. The results of research can be used in the study of Cauchy problems for partial differential equations with random initial conditions, for modeling real phenomena and processes in queueing theory, financial mathematics, physics, and others. On the other hand, the results can be used in teaching special courses on the theory of stochastic processes.