Multiscale Modeling &Simulation, Volume 18, Issue 1, Page 415-443, January 2020.
The Laplacian $\Delta$ is the infinitesimal generator of isotropic Brownian motion, being the limit process of normal diffusion, while the fractional Laplacian $\Delta^{\beta/2}$ serves as the infinitesimal generator of the limit process of isotropic Lévy process. Taking limit, in some sense, means that the operators can approximate the physical process well after sufficient long time. We introduce the nonlocal operators (being effective from the starting time), which describe the general processes undergoing anisotropic normal diffusion. For anomalous diffusion, we extend to the anisotropic fractional Laplacian $\Delta_m^{\beta/2}$ and the tempered one $\Delta_m^{\beta/2,\lambda}$ in $\mathbb{R}^n$. Their definitions are proved to be equivalent to an alternative one in Fourier space. Based on these new anisotropic diffusion operators, we further derive the deterministic governing equations of some interesting statistical observables of the very general jump processes with multiple internal states. Finally, we consider the associated initial and boundary value problems and prove their well-posedness of the Galerkin weak formulation in $\mathbb{R}^n$. To obtain the coercivity, we claim that the probability density function $Y$ should be nondegenerate.

中文翻译：

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正态和异常动力学的各向异性非局部扩散算子

2020年1月，《多尺度建模与仿真》，第18卷，第1期，第415-443页。
拉普拉斯算子\\ Delta $是各向同性布朗运动的无穷小生成器，是正态扩散的极限过程，而分数阶拉普拉斯算子\\ Delta ^ {\ beta / 2} $是各向同性Lévy的极限过程的无穷小生成器处理。从某种意义上说，采取极限意味着操作员在足够长的时间后可以很好地近似物理过程。我们介绍了非局部算子（从开始时间起生效），它描述了经历各向异性正态扩散的一般过程。对于异常扩散，我们扩展到各向异性分数拉普拉斯算子$ \ Delta_m ^ {\ beta / 2} $和在$ \ mathbb {R} ^ n $中回火的一个\\ Delta_m ^ {\ beta / 2，\ lambda} $ 。事实证明，它们的定义等同于傅立叶空间中的另一种定义。基于这些新的各向异性扩散算子，我们进一步推导了一些具有多种内部状态的非常普遍的跳跃过程的一些有趣的统计观测值的确定性控制方程。最后，我们考虑了相关的初始值和边值问题，并证明了$ \ mathbb {R} ^ n $中Galerkin弱公式的适定性。为了获得矫顽力，我们声称概率密度函数$ Y $应该是不退化的。