Vol. 31 (2025), no. 2 (Current Issue)

From an intuitive point of view, the multi-image construction procedure resembles the procedure of constructing the evolution scenario of some system in all possible reference frames, if we know its scenario of evolution in the given (fixed) reference frame. In the present paper it is investigated the problem of representation of a changeable set in the form of automultiimage that is in the form of the multi-image of some its reference frame. In particular we prove the necessary and sufficient condition for evolutionarily visible changeable set to be representable as automultiimage. Also using the last result we give the example of evolutionarily visible changeable set, which can not be represented as automultiimage.

In the review of the main modern positions of the moment problem, the importance of using the method of Berezansky Yu. M. - the expansion by generalized eigenvectors -- is discussed. This approach is currently the only correct one for solving the moment problem in different statements using the operator theory.

In the classical Gaussian analysis the Clark-Ocone formula can be written in the form \[ F=\mathbf EF+\int\mathbf E\big(\partial_t F|_{\mathcal F_t}\big)dW_t, \] where a function (a random variable) $F$ is square integrable with respect to the Gaussian measure and differentiable by Hida; $\mathbf E$ denotes the expectation; $\mathbf E\big(\circ|_{\mathcal F_t}\big)$---the conditional expectation with respect to the $\sigma$-algebra $\mathcal F_t$ that is generated by a Wiener process $W$ up to the point of time $t$; $\partial_\cdot F$ is the Hida derivative of $F$; $\int\circ (t)dW_t$ denotes the Itô stochastic integral over a Wiener process. This formula has many applications, in particular, in the stochastic analysis and in the financial mathematics.

In this paper we generalize the Clark-Ocone formula to the spaces $(\mathcal H_{-\tau})_{-q}$ of nonregular generalized functions in the Lévy white noise analysis. More exactly, we prove that any element of $(\mathcal H_{-\tau})_{-q}$ can be represented as a sum of its expectation and a result of stochastic integration over a Lévy process of some generalized function, and construct Clark-Ocone type formulas on $(\mathcal H_{-\tau})_{-q}$ and on its subsets.

In this review paper, we demonstrate that several classes of point processes in a locally compact Polish space $X$ appear as the joint spectral measure of a rigorously defined particle density of a representation of the canonical anticommutation relations (CAR) or the canonical commutation relations (CCR) in a Fock space. For these representations of the CAR/CCR, the vacuum state on the corresponding $*$-algebra is quasi-free. The classes of point process that arise in such a way include determinantal and permanental point processes.