Osypchuk M. Symmetrіc stable stochastic processes and theіr transformatіons

The thesis is devoted to investigating symmetric α-stable stochastic processes in Euclidean space associated with some class of pseudo-differential equations of parabolic type. A version of the theory of single-layer potential is constructed for those equations. The central point of that theory is the result being an analogy to the classical theorem on the jump of the co-normal derivative of a single-layer potential. Making use of this result, we construct fundamental solutions for the second and the third initial boundary-value problems for the equations under consideration. it turns out that in some cases those fundamental solutions are non-negative and they determine some Markov processes being certain transformations of a starting α-stable process. For example, a stochastic process in Euclidean space is constructed such that it describes an α-stable motion in that space with the sticky membrane located on a given surface. in others situations the constructed fundamental solutions do not determine any stochastic process but only “pseudo-process”. The results just described are expanded in the second chapter on the thesis. In the third chapter, the generator of an α-stable process is perturbed additively with a fractional gradient multiplied by a given vector field determined by bounded or locally unbounded and even generalized functions. The corresponding semigroups of operators are constructed and they turn out to be non-Markovian. it should be remarked than in the limit case of α=2, some of those transformations lead us to Markov processes, for example, a skew Brownian motion on a real line and some multidimensional analogues of it. The forth chapter is devoted to investigating some one-dimensional Markov processes related to a symmetric α-stable process. Those processes in the case of 1<α<2 turn out to be different, while they coincide each of the other one in the case of α=2. The last, fifth, chapter is devoted to some problems for Brownian motion (the case of α=2) or more general diffusion processes. in particular, an extreme problems consist in constructing drifts for a given Brownian motion that maximize the local time at the origin, minimize the time of the first hit in the origin have been solved. Other problems considered in Chapter 5 consist in investigating the limit behavior of the crossings number of a fixed level by a sequence of diffusion processes on a real line and of the local time at the origin of those processes. The local characteristics of the processes form this sequence do not converge to the corresponding ones of the limit processes in the case under consideration. The results of the theses possess the theoretical sense. They can be used in the theory of stochastic processes, in the theory of pseudo-differential equations, the stochastic models studying of natural and socioeconomic phenomena.