A generalization of operator stochastic integrals

Andrzej Łuczak

Un Cig Ji

Based on the quantum white noise theory, we formulate new types of anticipating quantum stochastic integrals by combining the Hitsuda–Skorokhod quantum stochastic integrals and the interactions between the integrands and the integrators. For our purpose, we prove various versions of analytic characterization theorems of symbols of white noise operators as more general cases. Also, several types of quantum (stochastic) gradients are formulated as continuous linear operators which are related to the quantum white noise derivatives and reflecting the interactions with the external noises. Finally, we formulate further systematical study of anticipating quantum stochastic integrals.

Chris Barnett, J. Martin Lindsay, Ivan F. Wilde

Quantum stochastic integrals have been constructed in various contexts [2, 3, 4, 5, 9] by adapting the construction of the classical L2-Itô-integral with respect to Brownian motion. Thus, the integral is first defined for simple integrands as a finite sum, then one establishes certain isometry relations or suitable bounds to allow the extension, by continuity, to more general integrands. The integrator is typically operator-valued, the integrand is vector-valued or operator-valued and the quantum stochastic integral is then given as a vector in a Hilbert space, or as an operator on the Hilbert space determined by its action on suitable vectors.

Horacio S. Wio

Stéphane Attal, R. L. Hudson

Zhiyuan Huang, Shunlong Luo

A nonlinear and stochastic analysis of free Bose field is established in the framework of white noise calculus. Wick algebra structure is introduced in the space of generalized operators generated by quantum white noise, some fundamental properties of the calculus based on the Wick algebra are investigated. As applications, quantum stochastic integrals and quantum stochastic differential equations are treated from the viewpoint of Wick calculus.