A theory of open systems based on stochastic differential equations

2014 ◽  
Vol 116 (4) ◽  
pp. 495-503 ◽  
Author(s):  
A. M. Basharov
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Open Systems

Analysis of the Closure Approximation for a Class of Stochastic Differential Equations

2017 ◽  
Vol 10 (2) ◽  
pp. 299-330
Author(s):  
Yunfeng Cai ◽  
Tiejun Li ◽  
Jiushu Shao ◽  
Zhiming Wang
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Open Systems ◽  
Numerical Study ◽  
Computational Cost ◽  
Nonlinear Problems ◽  
Exact Result ◽  
Moment Closure ◽  
Closure Approximation ◽  
The Moment

AbstractMotivated by the numerical study of spin-boson dynamics in quantum open systems, we present a convergence analysis of the closure approximation for a class of stochastic differential equations. We show that the naive Monte Carlo simulation of the system by direct temporal discretization is not feasible through variance analysis and numerical experiments. We also show that the Wiener chaos expansion exhibits very slow convergence and high computational cost. Though efficient and accurate, the rationale of the moment closure approach remains mysterious. We rigorously prove that the low moments in the moment closure approximation of the considered model are of exponential convergence to the exact result. It is further extended to more general nonlinear problems and applied to the original spin-boson model with similar structure.

On the Asymptotic Behaviour of Some Stochastic Differential Equations for Quantum States

2003 ◽  
Vol 06 (02) ◽  
pp. 223-243 ◽  
Author(s):  
Alberto Barchielli ◽  
Anna M. Paganoni
Keyword(s):  
Invariant Measure ◽  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Open Systems ◽  
General Theorem ◽  
Sufficient Conditions ◽  
A Posteriori ◽  
Finite Dimensional ◽  
Long Time ◽  
Pure States

In this article we study the long time behaviour of a class of stochastic differential equations introduced in the theory of measurements continuous in time for quantum open systems. Such equations give the time evolution of the a posteriori states for a system underlying a continual measurement. First of all we give conditions for the equation to preserve pure states and then, in the case of a finite-dimensional Hilbert space, we obtain sufficient conditions from which the stochastic equation for a posteriori states is ensured to map, for t → +∞, mixed states into pure ones. Finally we study existence and uniqueness of an invariant measure for the equations which preserve pure states. We give a general theorem for the purely diffusive case, again in the finite-dimensional case; we then apply it to some physical examples. For the purely jump case, an example is discussed in which the invariant measure exists and is unique.

Stochastic Differential Equations for Power Law Behaviors

2012 ◽  
Author(s):  
Bo Jiang ◽  
Roger Brockett ◽  
Weibo Gong ◽  
Don Towsley
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Power Law

scholarly journals Strong Solution Existence for a Class of Degenerate Stochastic Differential Equations

2020 ◽  
Vol 53 (2) ◽  
pp. 2220-2224
Author(s):  
William M. McEneaney ◽  
Hidehiro Kaise ◽  
Peter M. Dower ◽  
Ruobing Zhao
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Strong Solution ◽  
Solution Existence
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