The fractional material derivative appears as the fractional operator that governs the dynamics of the scaling limits of Lévy walks - a stochastic process that originates from the famous continuous-time random walks. It is usually defined as the Fourier–Laplace multiplier, therefore, it can be thought of as a pseudo-differential operator. In this paper, we show that there exists a local representation in time and space, pointwise, of the fractional material derivative. This allows us to define it on a space of locally integrable functions which is larger than the original one in which Fourier and Laplace transform exist as functions.

中文翻译：

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从 Lévy 步行到分数材料导数：逐点表示和数值方案


分数材料导数表现为控制 Lévy 游走缩放极限动态的分数算子，这是一种源自著名的连续时间随机游走的随机过程。它通常被定义为傅立叶-拉普拉斯乘子，因此，它可以被认为是伪微分算子。在本文中，我们证明了分数材料导数在时间和空间上存在逐点的局部表示。这使我们能够将其定义在一个局部可积函数的空间上，该空间比傅立叶和拉普拉斯变换作为函数存在的原始空间更大。