On the relation between the fractional Brownian motion and the fractional derivatives

The definition and simulation of fractional Brownian motion are considered from the point of view of a set of coherent fractional derivative definitions. To do it, two sets of fractional derivatives are considered: (a) the forward and backward and (b) the central derivatives, together with two representations: generalised difference and integral. It is shown that for these derivatives the corresponding autocorrelation functions have the same representations. The obtained results are used to define a fractional noise and, from it, the fractional Brownian motion. This is studied. The simulation problem is also considered.     

双边约束条件下随机van der Pol系统的分岔研究

利用随机光滑动力系统的Chebyshev正交多项式逼近方法,研究了双边约束条件下随机van der Pol系统的分岔现象.数值研究表明,双边约束随机van der Pol系统中不仅存在着丰富的倍周期分岔现象,还存在非光滑系统中所特有的擦边分岔.着重研究了随机非光滑系统中的擦边分岔,分析了随机因素对非光滑动力系统中擦边分岔的影响.研究表明,Chebyshev多项式逼近也是研究随机非光滑系统动力学行为的一种有效方法. 关键词： 非光滑动力系统 随机 van der Pol系统 擦边分岔 双边约束     

调频信号的单模激光线性模型随机共振

对单模激光增益模型的光强方程加入调频信号,用线性化近似方法计算了以δ函数形式关联的两白噪声驱动下光强的输出功率谱及信噪比. 结果表明,信噪比随抽运噪声和量子噪声强度的变化可出现典型随机共振,受调制信号振幅的影响,信噪比随载波信号频率和调制信号频率的变化出现抑制、单调上升、共振、抑制和共振等几种情况. 关键词： 抽运噪声 单模激光 随机共振 调频信号     

二维各向同性均匀随机介质中平面波的传播及其局域性

讨论了二维各向同性均匀随机介质中平面波的传播及其局域性.假设二维随机介质具有各向同性窄带高斯谱,运用随机泛函理论给出了二维介质中随机平面驻波的解析近似解及其幅度和相位的定量统计特性.计算结果证实了波在随机介质中传播时的局域性.将二维介质中的平面波和柱面波的传播特性作了对比.数值结果模拟了二维随机介质场及平面驻波,并验证了本文结果的正确性. 关键词： 随机介质 随机泛函 平面波 局域性     

Measuring fluorescence to track a quantum emitter's state: a theory review

We review the continuous monitoring of a qubit through its spontaneous emission, at an introductory level. Contemporary experiments have been able to collect the fluorescence of an artificial atom in a cavity and transmission line, and then make measurements of that emission to obtain diffusive quantum trajectories in the qubit's state. We give a straightforward theoretical overview of such scenarios, using a framework based on Kraus operators derived from a Bayesian update concept; we apply this flexible framework across common types of measurements including photodetection, homodyne, and heterodyne monitoring and illustrate its equivalence to the stochastic master equation formalism throughout. Special emphasis is given to homodyne (phase-sensitive) monitoring of fluorescence. The examples we develop are used to illustrate basic methods in quantum trajectories, but also to introduce some more advanced topics of contemporary interest, including the arrow of time in quantum measurement, and trajectories following optimal measurement records derived from a variational principle. The derivations we perform lead directly from the development of a simple model to an understanding of recent experimental results.     

Periodic solutions of a periodic stochastic human immunodeficiency virus model with distributed delay and cytotoxic T lymphocytes immune response

In this paper, a periodic stochastic human immunodeficiency virus (HIV) model with distributed delay and cytotoxic T lymphocytes (CTL) immune response is investigated. First, by It $ô$'s formula, we show that the solution with any positive initial value is global and positive. Then, by the stochastic comparison theorem, we obtain the sufficient conditions guaranteeing the existence and global attractivity of infection-free periodic solution. Furthermore, we discuss the existence of the infective periodic solution by Has'minskii theory. Finally, numerical examples are given to illustrate the results.     

Distribution dependent stochastic differential equations

Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications, distribution dependent stochastic differential equations (DDSDEs) have been intensively investigated. In this paper, we summarize some recent progresses in the study of DDSDEs, which include the correspondence of weak solutions and nonlinear Fokker-Planck equations, the well-posedness, regularity estimates, exponential ergodicity, long time large deviations, and comparison theorems.     

Discussion of “Virtual age,is it real?”

In this note, some point of views on virtual ages are presented in terms of the discussion paper written by Finkelstein and Cha, which include generalized stochastic order‐based virtual ages, system‐level virtual ages, virtual ages in Weibull distribution and repair degrees with virtual ages. Finally, some possible future researches on virtual ages are described.     

Analysis of a micro–macro acceleration method with minimum relative entropy moment matching

We analyse convergence of a micro–macro acceleration method for the simulation of stochastic differential equations with time-scale separation. The method alternates short bursts of path simulations with the extrapolation of macroscopic state variables forward in time. After extrapolation, a new microscopic state is constructed, consistent with the extrapolated macroscopic state, that minimises the perturbation caused by the extrapolation in a relative entropy sense. We study local errors and numerical stability of the method to prove its convergence to the full microscopic dynamics when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.