In this work, we compare Lévy space-time white noises and cylindrical Lévy processes. Lévy space-time white noises are defined as infinitely divisible independently scattered random measures and cylindrical Lévy processes are defined by means of the theory of cylindrical processes. It is shown that Lévy space-time white noises correspond to an entire sub-class of cylindrical Lévy processes, which is completely characterised by the characteristic functions of its members. We embed the Lévy space-time white noise, or the corresponding cylindrical Lévy process, in the space of general and tempered distributions. For the latter case, we show that this embedding is possible if and only if a certain integrability condition is satisfied. We establish that both embedded cylindrical processes are induced by genuine Lévy processes in the space of general or tempered distributions. We complete the picture by establishing Lévy space-time white noise as the weak derivative of Lévy additive sheets.

中文翻译：

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建模 Lévy 时空白噪声

在这项工作中，我们比较了 Lévy 时空白噪声和圆柱 Lévy 过程。Lévy 时空白噪声被定义为无限可分的独立散射随机测度，圆柱 Lévy 过程是通过圆柱过程理论定义的。结果表明，Lévy 时空白噪声对应于圆柱 Lévy 过程的整个子类，其完全由其成员的特征函数表征。我们将 Lévy 时空白噪声或相应的圆柱 Lévy 过程嵌入到一般分布和调和分布的空间中。对于后一种情况，我们表明当且仅当满足某个可积性条件时，这种嵌入是可能的。我们确定这两个嵌入式圆柱过程都是由一般或调和分布空间中的真正 Lévy 过程引起的。我们通过建立 Lévy 时空白噪声作为 Lévy 加法片的弱导数来完成图片。