Dong Myung Chung

Transforms on white noise functionals with their applications to Cauchy problems

A generalized Laplacian Δ G (K) is defined as a continuous linear operator acting on the space of test white noise functionals. Operator-parameter - and -transforms on white noise functionals are introduced and then prove a characterization theorem for and -transforms in terms of the coordinate differential operator and the coordinate multiplication. As an application, we investigate the existence and uniqueness of solution of the Cauchy problem for the heat equation associated with Δ G (K)

QUANTUM STOCHASTIC ANALYSIS VIA WHITE NOISE OPERATORS IN WEIGHTED FOCK SPACE

White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Ito 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 Ito 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.

CAUCHY PROBLEMS ASSOCIATED WITH THE LÉVY LAPLACIAN IN WHITE NOISE ANALYSIS

In this paper we shall discuss the existence and the uniqueness of solutions of the heat type equation and the wave type equation associated with the Levy Laplacian acting on a domain in the space of generalized white noise functionals.

Higher Powers of Quantum White Noises in Terms of Integral Kernel Operators

A rigorous mathematical formulation of higher powers of quantum white noises is given on the basis of the most recent theory of white noise distributions due to Cochran, Kuo and Sengupta. The renormalized quantum Ito formula due to Accardi, Lu and Volovich is derived from the renormalized product formula based on integral kernel operators on white noise functions. During the discussion, the analytic characterization of operator symbols and the expansion theorem for a white noise operator in terms of integral kernel operators are established.

Notes on a C~0-group generated by the Levy Laplacian

In this paper we shall give some results on a C 0 -group generated by the Levy Laplacian and operators approximating that group in the space L(E) of continuous linear operators defined on a certain locally convex space E in (S) * .

Some cauchy problems in white noise analysis and associated semigroups of operators

Let (E)β and be the space of test and generalized white noise functionals, respectively. It is proved that for each positive integer k we construct explicitly one-parameter semigroup and cosine family of operators on an appropriate (E)β space of which infinitesimal generator is . As an application, we will prove the existence and uniqueness of solutions of the Cauchy problems for the first and second order differential equations associated with the operator . Here △ G and Nare the Gross Laplacian and the number operator on (E)β , respectively. Also, we investigate the existence and uniqueness of solutions of the Cauchy problems associated with the operator is the adjoint operator of △ G

Evaluation formulas for conditional abstract wiener integrals

In this paper we consider abstract Wiener space version of conditional Wiener integrals and establish formulas for evaluating conditional abstract Wiener integrals for various classes of functions on an abstract Wiener space. We then apply our formulas to evaluate certain Wiener integrals and conditional Wiener and Yeh-Wiener integrals

CONDITIONAL INTEGRALS ON ABSTRACT WIENER AND HILBERT SPACES WITH APPLICATION TO FEYNMAN INTEGRALS

In this paper, we define conditional integrals on ab- stract Wiener and Hilbert spaces and then obtain a formula for evaluating the integrals. We use this formula to establish the ex- istence of conditional Feynman integrals for the classesq (B) and � q (H) of functions on abstract Wiener and Hilbert spaces and then specialize this result to provide the fundamental solution to the Schrodinger equation with the forced harmonic oscillator.

Poisson Equations Associated with Differential Second Quantization Operators in White Noise Analysis

In this paper we shall show the heredity of a differentiable one-parameter semigroup under the second quantization and then discuss the resolvent of the differential second quantization operator and the potentials of test white noise functionals. As an application, we shall investigate the existence of solutions of the Poisson-type equations associated with differential second quantization operators as well as operators similar to differential second quantization operators.