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Journal of Lanzhou University of Technology

Adaptive solutions to quantum stochastic differential equations

Guo Jin-sheng

Published Jan 1, 2013• 0 citations
OpenAlex

Abstract

A class of generalized operator Wick filtration with R as its index set was constructed within the analysis framework of biparametric white noise.Fundamental properties of this Wick filtration were discussed.And a notion of adaptability of quantum stochastic processes in terms of generalized operators to the Wick filtration was introduced.The adaptive solution of a class of quantum stochastic differential equations to this Wick filtration was explored.

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A class of generalized operator Wick filtration with R as its index set was constructed within the analysis framework of biparametric white noise.Fundamental properties of this Wick filtration were discussed.And a notion of adaptability of quantum stochastic processes in terms of generalized operators to the Wick filtration was introduced.The adaptive solution of a class of quantum stochastic differential equations to this Wick filtration was explored.
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.
In this paper we introduce the quantum white noise (QWN) conservation operator N Q acting on nuclear algebra of white noise operators) endowed with the Wick product.Similarly to the classical case, we give a useful integral representation in terms of the QWNderivatives {D - t , D + t ; t ∈ R} for the QWN-conservation operator from which it follows that the QWN-conservation operator is a Wick derivation.Via this property, a relation with the Cauchy problem associated to the QWN-conservation operator and the Wick differential equation is worked out.