On Wick calculus on spaces of nonregular generalized functions of Levy white noise analysis
Journal Title: Карпатські математичні публікації - Year 2018, Vol 10, Issue 1
Abstract
Development of a theory of test and generalized functions depending on infinitely many variables is an important and actual problem, which is stipulated by requirements of physics and mathematics. One of successful approaches to building of such a theory consists in introduction of spaces of the above-mentioned functions in such a way that the dual pairing between test and generalized functions is generated by integration with respect to some probability measure. First it was the Gaussian measure, then it were realized numerous generalizations. In particular, important results can be obtained if one uses the Levy white noise measure, the corresponding theory is called the Levy white noise analysis. In the Gaussian case one can construct spaces of test and generalized functions and introduce some important operators (e.g., stochastic integrals and derivatives) on these spaces by means of a so-called chaotic representation property (CRP): roughly speaking, any square integrable random variable can be decomposed in a series of repeated Itos stochastic integrals from nonrandom functions. In the Levy analysis there is no the CRP, but there are different generalizations of this property. In this paper we deal with one of the most useful and challenging generalizations of the CRP in the Levy analysis, which is proposed by E.W. Lytvynov, and with corresponding spaces of nonregular generalized functions. The goal of the paper is to introduce a natural product (a Wick product) on these spaces, and to study some related topics. Main results are theorems about properties of the Wick product and of Wick versions of holomorphic functions. In particular, we prove that an operator of stochastic differentiation satisfies the Leibniz rule with respect to the Wick multiplication. In addition we show that the Wick products and the Wick versions of holomorphic functions, defined on the spaces of regular and nonregular generalized functions, constructed by means of Lytvynov's generalization of the CRP, coincide on intersections of these spaces. Our research is a contribution in a further development of the Levy white noise analysis.
Authors and Affiliations
N. A. Kachanovsky
Keywords
Related Articles
- EP ID EP532782
- DOI 10.15330/cmp.10.1.114-132
- Views 68
- Downloads 0
On the convergence criterion for branched continued fractions with independent variables
In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an ef...
Faithful group actions and Schreier graphs
Each action of a finitely generated group on a set uniquely defines a labelled directed graph called the Schreier graph of the action. Schreier graphs are used mainly as a tool to establish geometrical and dynamical pro...
On a necessary condition for Lp (0<p<1) -convergence (upper boundedness) of trigonometric series
In this paper we prove that the condition ∑2nk=[n2]λk(p)(|n−k|+1)2−p=o(1)(=O(1)), is a necessary condition for the Lp(0<p<1)-convergence (upper boundedness) of a trigonometric series. Precisely, the results extend some r...
Separating polynomials and uniform analytical and separating functions
We present basic results of the theory of separating polynomials and uniformly analytic and separating functions on separable real Banach spaces. We consider basic properties of separating polynomials and uniformly analy...
Properties of distance spaces with power triangle inequalities
Metric spaces provide a framework for analysis and have several very useful properties. Many of these properties follow in part from the triangle inequality. However, there are several applications in which the triangle...