Interconnection between Wick multiplication and integration on spaces of nonregular generalized functions in the Lévy white noise analysis

We deal with spaces of nonregular generalized functions in the Lévy white noise analysis, which are constructed using Lytvynov's generalization of a chaotic representation property. Our aim is to describe a relationship between Wick multiplication and integration on these spaces. More exactly, we show that when employing the Wick multiplication, it is possible to take a time-independent multiplier out of the sign of an extended stochastic integral; establish an analog of this result for a Pettis integral (a weak integral); and prove a theorem about a representation of the extended stochastic integral via the Pettis integral from the Wick product of the original integrand by a Lévy white noise. As examples of an application of our results, we consider some stochastic equations with Wick type nonlinearities.

Approximation of a free Poisson process by systems of freely independent particles

Let [Formula: see text] be a non-atomic, infinite Radon measure on [Formula: see text], for example, [Formula: see text] where [Formula: see text]. We consider a system of freely independent particles [Formula: see text] in a bounded set [Formula: see text], where each particle [Formula: see text] has distribution [Formula: see text] on [Formula: see text] and the number of particles, [Formula: see text], is random and has Poisson distribution with parameter [Formula: see text]. If the particles were classically independent rather than freely independent, this particle system would be the restriction to [Formula: see text] of the Poisson point process on [Formula: see text] with intensity measure [Formula: see text]. In the case of free independence, this particle system is not the restriction of the free Poisson process on [Formula: see text] with intensity measure [Formula: see text]. Nevertheless, we prove that this is true in an approximative sense: if bounded sets [Formula: see text] ([Formula: see text]) are such that [Formula: see text] and [Formula: see text], then the corresponding particle system in [Formula: see text] converges (as [Formula: see text]) to the free Poisson process on [Formula: see text] with intensity measure [Formula: see text]. We also prove the following [Formula: see text]-limit: Let [Formula: see text] be a deterministic sequence of natural numbers such that [Formula: see text]. Then the system of [Formula: see text] freely independent particles in [Formula: see text] converges (as [Formula: see text]) to the free Poisson process. We finally extend these results to the case of a free Lévy white noise (in particular, a free Lévy process) without free Gaussian part.

On Wick calculus on spaces of nonregular generalized functions of Levy white noise analysis

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.