The investigation, results of which are expounded in the thesis, refers to an infinite dimensional analysis—a quite wide area of modern mathematics, in which one studies, in particular, different spaces of functions and generalized functions of infinite many variables (i.e., of functions and generalized functions with arguments from infinite dimensional spaces).In the thesis we introduce and investigate operators of stochastic differentiation on spaces of regular test and generalized functions of the Lèvy white noise analysis–one of the most sought-generalizations of the classical Gaussian white noise analysis, and develop a Wick calculus on the mentioned spaces of regular generalized functions.Operators of stochastic differentiation, which are closely related with the extended stochastic integral and with the Hida stochastic derivative, play an important role in the Gaussian white noise analysis. In particular, these operators can be used in order to study properties of the extended stochastic integral and properties of solutions of certain stochastic integral and differential equations.Another important object in the Gaussian white noise analysis is a Wick calculus on the spaces of generalized functions, i.e., a theory, which studies natural analogues of the pointwise product (a so-called Wick product) and of holomorphic functions (so-called Wick versions of holomorphic functions) on the mentioned spaces, as well as stochastic integral and differential equations with the Wick product and with the Wick versions of holomorphic functions (so-called stochastic equations with Wick-type nonlinearities). The mentioned equations have applications, in particular, in the stochastic analysis and in the mathematical physics. It is worth noting that in order to study some properties of solutions of such equations, one can use the operators of stochastic differentiation (in particular, one can use the fact that the operator of stochastic differentiation of first order is the differentiation with respect to the Wick product, i.e., this operator satisfies the Leibniz rule).The thesis is organized as follows. First we did a review of the literature and of known results on the subject of the work; define the Lèvy white noise measure and describe related topics; describe Lytvynov’s (2003) generalization of the chaotic representation property; introduce the parametrized Kondratiev-type spaces of regular test and generalized functions (that are positive and negative spaces of a so-called parametrized regular rigging of the space of square integrable with respect to the Lèvy white noise measure functions respectively) in terms of the mentioned generalization of the chaotic representation property, and present decompositions of elements belonging to these spaces by natural orthogonal bases; describe constructions of the extended Skorohod stochastic integral and of the Hida stochastic derivative on the mentioned spaces of regular test and generalized functions.Then we introduce and study in detail operators of stochastic differentiation on the spaces of regular test and generalized functions of the Lèvy white noise analysis. Separately we consider the cases, in which the mentioned operators are bounded and unbounded. After this, by analogy with the Gaussian white noise analysis, we construct elements of a Wick calculus on the spaces of regular generalized functions. In particular, we give definitions and study properties of a Wick product and of Wick versions of holomorphic functions; establish that the operator of stochastic differentiation of first order is a differentiation (satisfies the Leibniz rule) with respect to the Wick multiplication; show that if one uses the Wick multiplication instead of the pointwise multiplication, then it is possible to take the time-independent multiplier out of the sign of the stochastic integral; formulate and prove a theorem about presentation of the extended stochastic integral via a formal Pettis integral from the Wick product of the initial integrand and the Lèvy white noise. These results are a basis for further development of the Lèvy white noise analysis, and can be used, in particular, for solving of stochastic integral and differential equations with Wick-type nonlinearities on the spaces of regular generalized functions. It is worth noting that, as in the Gaussian analysis, the mentioned equations can be used for modelling of different physical processes.
