# Kachanovskii N. A.

### Elements of a non-gaussian analysis on the spaces of functions of infinitely many variables

Ukr. Mat. Zh. - 2010. - 62, № 9. - pp. 1220–1246

We present a review of some results of the non-Gaussian analysis in the biorthogonal approach and consider elements of the analysis associated with the generalized Meixner measure. The main objects of our interest are stochastic integrals, operators of stochastic differentiation, elements of theWick calculus, and related topics.

### Stochastic integral of Hitsuda–Skorokhod type on the extended Fock space

Kachanovskii N. A., Tesko V. A.

Ukr. Mat. Zh. - 2009. - 61, № 6. - pp. 733-764

We review some recent results related to stochastic integrals of the Hitsuda–Skorokhod type acting on the extended Fock space and its riggings.

### Generalized stochastic derivatives on spaces of nonregular generalized functions of Meixner white noise

Ukr. Mat. Zh. - 2008. - 60, № 6. - pp. 737–758

We introduce and study generalized stochastic derivatives on Kondratiev-type spaces of nonregular generalized functions of Meixner white noise. Properties of these derivatives are quite analogous to properties of stochastic derivatives in the Gaussian analysis. As an example, we calculate the generalized stochastic derivative of a solution of a stochastic equation with Wick-type nonlinearity.

### Extended Stochastic Integral and Wick Calculus on Spaces of Regular Generalized Functions Connected with Gamma Measure

Ukr. Mat. Zh. - 2005. - 57, № 8. - pp. 1030–1057

We introduce and study an extended stochastic integral, a Wick product, and Wick versions of holomorphic functions on Kondrat'ev-type spaces of regular generalized functions. These spaces are connected with the Gamma measure on a certain generalization of the Schwartz distribution space \(S'\). As examples, we consider stochastic equations with Wick-type nonlinearity.

### On Orthogonal Appell-Like Polynomials in Non-Gaussian Analysis

Kachanovskii N. A., Kalyuzhnyi A. A.

Ukr. Mat. Zh. - 2001. - 53, № 7. - pp. 892-907

We study an example of the construction of a non-Gaussian analysis using orthogonal generalized Appell-like polynomials with the generating function $$\frac{1}{{\sqrt {1 - 2a{\lambda + \lambda }^{2}} } }\cos \left( {\sqrt x \frac{1}{2}\int\limits_{0}^{\lambda } {\frac{{du}}{{\sqrt {u - 2au^2 + u^3 } }}} } \right),\quad a >1,$$ in the model one-dimensional case. The main results are a detailed intrinsic description of spaces of test functions, a description of generalized translation operators, and the investigation of integral *C*- and *S*-transformations.

### Pseudodifferential equations and a generalized translation operator in non-gaussian infinite-dimensional analysis

Ukr. Mat. Zh. - 1999. - 51, № 10. - pp. 1334–1341

Pseudodifferential equations of the form $v(D_{\chi})y = f$ (where $v$ is a function holomorphic at zero and $D_{\chi}$ is a pseudodifferential operator) are studied on spaces of test functions of non-Gaussian infinite-dimensional analysis. The results obtained are applied to construct a generalized translation operator $T^{\chi}_y = \chi(\langle y, D_{\chi}\rangle)$ the already mentioned spaces and to study its properties. In particular, the associativity, the commutativity, and another properties of $T^{\chi}_y$ which are analogs of the classical properties of a generalized translation operator.

### Dual Appell system and Kondrat’ev spaces in analysis on Schwartz spaces

Ukr. Mat. Zh. - 1997. - 49, № 4. - pp. 527–534

We study the biorthogonal Appell system and Kondrat’ev spaces in the case where the parameter of a μ-exponential is perturbed by holomorphic invertible functions. The results obtained are applied to the investigation of pseudodifferential equations.