scholarly journals Quantum stochastic process associated with quantum Lévy Laplacian

2007 ◽  
Vol 1 (2) ◽  
Author(s):  
Un Cig Ji ◽  
Habib Ouerdiane ◽  
Kimiaki Saitô
Keyword(s):  
Stochastic Process ◽  
Quantum Stochastic Process ◽  
Lévy Laplacian

A SIMILARITY BETWEEN THE GROSS LAPLACIAN AND THE LÉVY LAPLACIAN

2007 ◽  
Vol 10 (02) ◽  
pp. 261-276 ◽  
Author(s):  
UN CIG JI ◽  
KIMIAKI SAITÔ
Keyword(s):  
Stochastic Process ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Existence And Uniqueness ◽  
Infinitesimal Generator ◽  
Separable Hilbert Space ◽  
Fock Spaces ◽  
Parameter Group ◽  
Infinite Dimensional ◽  
Lévy Laplacian

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.

SYMMETRIC DIFFERENTIATION AND HAMILTONIAN OF A QUANTUM STOCHASTIC PROCESS

2005 ◽  
Vol 08 (01) ◽  
pp. 73-116 ◽  
Author(s):  
WILHELM VON WALDENFELS
Keyword(s):  
Differential Equation ◽  
Stochastic Process ◽  
Stochastic Calculus ◽  
One Dimension ◽  
Time Shift ◽  
Quantum Stochastic Calculus ◽  
Quantum Stochastic Differential Equation ◽  
Differentiation Formula ◽  
Quantum Stochastic Process ◽  
Creation Operators

The main object of this paper is to establish the Hamiltonian of the Hudson–Parthasarathy quantum stochastic differential equation [Formula: see text] As its solution forms a cocycle with respect to the time shift, its product with the time shift establishes a strongly continuous one-parameter unitary group W(t)= exp (-itH) and H is called its Hamiltonian. Our main result is that H is the closure of the restriction of the singular operator [Formula: see text] on an appropriate domain that shall be explicitly described. The symmetric differentiation [Formula: see text] is a generalization of the generator of the time shift, [Formula: see text] and [Formula: see text] are annihilation and creation operators and [Formula: see text] is the symmetric Dirac δ-function. In one dimension the symmetric differentiation might be used to establish a quantum stochastic calculus without Ito term.

DIAGONALIZATION OF THE LÉVY LAPLACIAN AND RELATED STABLE PROCESSES

2002 ◽  
Vol 05 (03) ◽  
pp. 317-331 ◽  
Author(s):  
HUI-HSIUNG KUO ◽  
NOBUAKI OBATA ◽  
KIMIAKI SAITÔ
Keyword(s):  
Stochastic Process ◽  
Continuous Semigroup ◽  
Stable Process ◽  
Arbitrary Real Number ◽  
Stable Processes ◽  
Direct Integral ◽  
Coordinate Change ◽  
One Dimensional ◽  
Infinite Dimensional ◽  
Lévy Laplacian

Eigenfunctions of the Lévy Laplacian with an arbitrary real number as an eigenvalue are constructed by means of a coordinate change of white noise distributions. The Lévy Laplacian is diagonalized on the direct integral Hilbert space of such eigenfunctions and the corresponding equi-continuous semigroup is obtained. Moreover, an infinite dimensional stochastic process related to the Lévy Laplacian is constructed from a one-dimensional stable process.

scholarly journals QUANTUM MARKOVIAN APPROXIMATIONS FOR FERMIONIC RESERVOIRS

2005 ◽  
Vol 08 (03) ◽  
pp. 453-471 ◽  
Author(s):  
JOHN GOUGH ◽  
ANDREI SOBOLEV
Keyword(s):  
Stochastic Process ◽  
Central Limit ◽  
Dynamical Evolution ◽  
Quantum Stochastic Process ◽  
Markovian Approximations ◽  
The Creation ◽  
Functional Central Limit ◽  
Markovian Limit

We establish a quantum functional central limit for the dynamics of a system coupled to a fermionic bath with a general interaction linear in the creation, annihilation and scattering of the bath reservoir. Following a quantum Markovian limit, we realize the open dynamical evolution of the system as an adapted quantum stochastic process driven by fermionic noise.

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