scholarly journals Reconstruction theorem for a quantum stochastic process

1985 ◽  
Vol 62 (3) ◽  
pp. 275-289 ◽  
Author(s):  
V. P. Belavkin
Keyword(s):  
Stochastic Process ◽  
Quantum Stochastic Process ◽  
Reconstruction Theorem

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.

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.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

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