Narrow Results

In this paper, we construct a quantum analogue of a white noise delta function. For our main purpose, we first discuss several white noise delta functions, specially, an infinite-dimensional analogue of Donsker’s delta function. Then as a general setting, we construct white noise delta functions on white noise test functionals and we derive a Cauchy problem whose solution is given by the white noise delta function. Also, we prove that the white noise delta function is an eigenvector of the exotic Laplacian. By applying the canonical topological isomorphisms between the spaces of white noise operators and the spaces of white noise functionals, we formulate a quantum analogue of the white noise delta function which satisfies a quantum white noise differential equation associated with the quantum Volterra Laplacian. Finally, we prove that quantum analogue of the white noise delta function is an eigenoperator of the quantum exotic Laplacian.

White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.

We revisit a CKS-space from the viewpoint of standard setup of white noise calculus and prove a general characterization theorem for white noise operators from a CKS-space into another CKS-space. The usual characterizations so far obtained for white noise distributions, white noise test functions and white noise operators are all consequences of the unified characterization theorem.

A stochastic integral representation in terms of generalized integral kernel operator is proved for a wide class of quantum martingales which includes regular martingales and the martingales determined by the second quantization of integral operators containing singular kernels.

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

In this paper, we give a decomposition of the space of tempered distributions by the Cesàro norm, and for any we construct directly from the exotic trace an infinite dimensional separable Hilbert space H_c,2a-1 on which the exotic trace plays the role as the usual trace. This implies that the Exotic Laplacian coincides with the Volterra–Gross Laplacian in the Boson Fock space Γ(H_c,2a-1) over the Hilbert space H_c,2a-1. Finally we construct the Brownian motion naturally associated to the Exotic Laplacian of order 2a-1 and we find an explicit expression for the associated heat semigroup.

In this paper, we give a relationship between the exotic Laplacians and the Lévy Laplacians in terms of the higher-order derivatives of white noise by introducing a bijective and continuous linear operator acting on white noise functionals. Moreover, we study a relationship between exotic Laplacians, acting on higher-order singular functionals, each other in terms of the constructed operator.

We consider a family of infinite dimensional Laplace operators which contains the classical Lévy–Laplacian. We prove a representation of these operators as a quadratic functions of quantum stochastic processes. Particularly, for the classical Lévy–Laplacian, the following formula is proved: Δ_L = lim_ε→0 ∫_‖s-t‖<ε b_sb_tdsdt, where b_t is the annihilation process.

We revisit the analytic characterization theorem for S-transform of infinite dimensional distributions. Then we prove that the nuclearity of the space of test white noise functionals is a necessary condition for the characterization of the S-transform in terms of analytic and growth conditions.

In this paper, we prove a large deviation principle for the background field in prequantum statistical field model. We show a number of examples by choosing a specific random field in our model.

The classical maximum principle for optimal stochastic control states that if a control $û$ is optimal, then the corresponding Hamiltonian has a maximum at $u=û$. The first proofs for this result assumed that the control did not enter the diffusion coefficient. Moreover, it was assumed that there were no jumps in the system. Subsequently, it was discovered by Shige Peng (still assuming no jumps) that one could also allow the diffusion coefficient to depend on the control, provided that the corresponding adjoint backward stochastic differential equation (BSDE) for the first-order derivative was extended to include an extra BSDE for the second-order derivatives. In this paper, we present an alternative approach based on Hida–Malliavin calculus and white noise theory. This enables us to handle the general case with jumps, allowing both the diffusion coefficient and the jump coefficient to depend on the control, and we do not need the extra BSDE with second-order derivatives. The result is illustrated by an example of a constrained linear-quadratic optimal control.

We study the analytic characterization of $S$-transform in a general setting of white noise functionals. Then, the measurability condition of the norms generating the underlining locally convex space is a necessary and sufficient condition for the analytic characterization of the $S$-transform in terms of analytic and growth conditions.

An integral representation of Hilbert–Schmidt operators on Boson Fock space is proved with explicit forms of integrands which is a quantum version of (classical) Clark–Haussmann–Ocone formula.

Observing that distribution theory offers an extension of the concept of integration, we review a framework in which the Feynman integral becomes mathematically meaningful for large classes of interaction potentials. We present some examples and open problems.

We introduce a new noise P′(υ,λ), with a space parameter υ and the intensity λ noting that the characteristic of Poisson type process depends on the intensity. A new noise P′(υ,λ) is different with the time derivative of Poisson process P(t). In this paper, the invariance of the intensity λ under the transformations : dilation and reflection is discussed. In addition, it is interesting to see how the intensity changes when the exponent of a stable process changes to its reciprocal.