BOUNDED OPERATORS OF STOCHASTIC DIFFERENTIATION ON SPACES OF NONREGULAR GENERALIZED FUNCTIONS IN THE L É VY WHITE NOISE ANALYSIS

Background . Operators of stochastic differentiation play an important role in the Gaussian white noise analysis. In particular, they can be used in order to study properties of the extended stochastic integral and of solutions of normally ordered stochastic equations. Although the Gaussian analysis is a developed theory with numerous applications, in problems of mathematics not only Gaussian random processes arise. In particular, an important role in modern re-searches belongs to L é vy processes. So, it is necessary to develop a L é vy analysis, including the theory of operators of stochastic differentiation. Objective. During recent years the operators of stochastic differentiation were introduced and studied, in particular, on spaces of regular test and generalized functions and on spaces of nonregular test functions of the L é vy analysis. In this paper, we make the next step: introduce and study such operators on spaces of nonregular generalized functions. Methods. We use, in particular, the theory of Hilbert equipments and Lytvynov’s generalization of the chaotic representation property. Results. The main result is a theorem about properties of operators of stochastic differentiation. Conclusions. The operators of stochastic differentiation are considered on the spaces of nonregular generalized functions of the L é vy white noise analysis. This can be interpreted as a contribution in a further development of the L é vy analysis. Applications of the introduced operators are quite analogous to the applications of the corresponding operators in the Gaussian analysis.


Denote
: Lévy process, i.e., a random process on R  with stationary independent increments and such that 0 0 L  (see, e.g., [1] for details), without Gaussian part and drift. In [2] the extended Skorohod stochastic integral with respect to L and the corresponding Hida stochastic derivative on the space of square integrable random variables 2 ( ) L were constructed in terms of Lytvynov's generalization of the chaotic representation property (CRP) [3], some properties of these operators were established; and it was shown that the above-mentioned integral coincides with the well-known extended stochastic integral with respect to a Lévy process, constructed in terms of Itô's generalization of the CRP [4] (see, e.g., [5,6]). In [7,8] the stochastic integral and derivative were extended to spaces of test and generalized functions that belong to riggings of 2 ( ) L , this gives a possibility to extend an area of their possible applications. Together with the mentioned operators, it is natural to introduce and to study so-called operators of stochastic differentiation in the Lévy white noise analysis, by analogy with the Gaussian analysis [9,10]. Such operators are closely related with the extended sto-chastic integral with respect to a Lévy process and with the corresponding Hida stochastic derivative and, by analogy with the "classical case", can be used, in particular, in order to study properties of the extended stochastic integral and properties of solutions of so-called normally ordered stochastic equations. In [11,12] the operators of stochastic differentiation on spaces belonging to a so-called regular parametrized rigging of 2 ( ) L [7] were introduced and studied. But, in connection with some problems of the stochastic analysis, sometimes it can be necessary to consider another, a so-called nonregular rigging of 2 ( ) L [7] and various operators on spaces (of nonregular test and generalized functions) belonging to this rigging. In [13] operators of stochastic differentiation were introduced and studied on the spaces of nonregular test functions of the Lévy white noise analysis. In particular, it was shown that these operators are the restrictions to the above-mentioned spaces of the corresponding operators on 2 ( ) L . The next natural step consists in introduction and study of operators of stochastic differentiation on the spaces of nonregular generalized functions. But, unfortunately, the operators of stochastic differentiation on 2 ( ) L (in the same way as the Hida stochastic derivative) cannot be naturally continued to the abovementioned spaces. Nevertheless, one can introduce on these spaces linear operators with properties quite analogous to properties of the operators of stochastic differentiation. These linear operators will be called the operators of stochastic differentiation on the spaces of nonregular generalized functions.
In the present paper we introduce these operators and establish some their properties.

Problem definition
The aim of this paper is to introduce the operators of stochastic differentiation on the spaces of nonregular generalized functions of the Lévy white noise analysis; and to establish some properties of these operators.

Preliminaries
In this paper we deal with a real-valued locally square integrable Lévy process L on R  without Gaussian part and drift. As is well known, the characteristic function of such a process is where  is the Lévy measure of L, E denotes the expectation. We assume that  is a Radon measure whose support contains an infinite number of points, Let us define a measure of the white noise of L . Let D denote the set of all real-valued infnitedifferentiable functions on R  with compact supports. As is well known, D can be endowed by the projective limit topology generated by some Sobolev spaces (more details are given below, a detailed presentation is given in, e.g., [14] is called the Lévy white noise measure. Denote ( ) [2,3,5,6] that , : and below 1 A denotes the indicator of a set A ). It follows from (1) and (2) that can be identified with a Lévy process L. Consider Lytvynov's generalization of the CRP (see [3] for details). Denote by   the symmetric tensor product. For Denote by : , : : ,  weighted by the function 2  (e.g., [14]). It is well known that and any formal series (6) with finite norm (7) (as is well known (e.g., [14]),  (the symmetrization operator), 1 is the identity operator. It is shown in [13] that this integral is an extension of the extended Skorohod stochastic integral with respect to a Lévy process L .
Unfortunately, the extended stochastic integral with respect to a Lévy process cannot be naturally restricted to the spaces of nonregular test functions. More precisely, for ( ) ( )  operator). The well-posedness of this definition is proved in [13].
Finally, as is well known, an important role in the Lévy white noise analysis belongs to the Hida stochastic derivative, which is the adjoint operator of the extended stochastic integral. In terms of Lytvynov's generalization of the CRP this derivative is considered on 2 ( ) L [2], on the spaces of regular test and generalized functions [11,12] and on the spaces of nonregular test functions [7,13]. But, unfortunately, this operator has no a natural extension to the spaces of nonregular generalized functions. Nevertheless, one can define natural analogs of the Hida stochastic derivative on these spaces as operators adjoint to I. are the kernels from decomposition (6) for F.

Operators of stochastic differentiation
As we said above, the operators of stochastic differentiation on 2 ( ) L [11,12] cannot be naturally continued to the spaces of nonregular generalized functions (because the kernels from decompositions (6) for elements of ( ) q H   belong to the spaces wider than ). Nevertheless, one can introduce on these spaces natural analogs of the above-mentioned operators. We begin from a preparation. Let   (7) and (13)