Initially developed in the framework of quantum stochastic calculus, the main equations of quantum stochastic filtering were later on derived as the limits of Markov models of discrete measurements under appropriate scaling. In many branches of modern physics it became popular to extend random walk modeling to the continuous time random walk (CTRW) modeling, where the time between discrete events is taken to be non-exponential. In the present paper we apply the CTRW modeling to the continuous quantum measurements yielding the new fractional in time evolution equations of quantum filtering and thus new fractional equations of quantum mechanics of open systems. The related quantum control problems and games turn out to be described by the fractional Hamilton-Jacobi-Bellman (HJB) equations on Riemannian manifolds. By-passing we provide a full derivation of the standard quantum filtering equations, in a modified way as compared with existing texts, which (i) provides explicit rates of convergence (that are not available via the tightness of martingales approach developed previously) and (ii) allows for the direct applications of the basic results of CTRWs to deduce the final fractional filtering equations.
Random walk models approximating symmetric space-fractional diffusion processes
对于对称的情况下的空间分数扩散过程（其基本的分析理论已在1952年由Feller通过反演里斯位势算子），我们提出了三个随机游动模型离散的空间和时间。我们表明，适当缩放过渡到消失的空间和时间的步骤，这些模型收敛于相应的时间参数化的稳定概率分布的分布。最后，我们详细分析了一个模型，离散的时间，但连续的空间，最近提出的Chechkin和Gonchar。关于里斯位势算子$I_0^α$的反演，我们指出，用解析延拓，它的一般超奇异积分表示对$α= 1. In our Section 2 we have shown that the corresponding hyper-singular representation for the inverse operator $ D_0 ^α $ can be obtained also in the critical (often excluded) case $ α= 1$是不成立的
Dynamical Localization for General Scattering Quantum Walks
Univ. Grenoble Alpes, CNRS, Institut Fourier（格勒诺布尔阿尔卑斯大学，法国国家科学研究中心，傅里叶研究所）；TU Munich（慕尼黑工业大学）；Munich Center for Quantum Science and Technology（慕尼黑量子科学与技术中心）
A Fractional Variational Approach to Spectral Filtering Using the Fourier Transform
Laboratory of Mathematical-Chemistry, Institute of Chemistry, Universidade Federal de Alfenas (UNIFAL)（数学化学实验室，化学学院，联邦阿尔费纳斯大学）；Institute of Exact Science, Universidade Federal de Alfenas (UNIFAL)（精确科学学院，联邦阿尔费纳斯大学）
