We introduce L\'evy-driven causal CARMA random fields on $\mathbb{R}^d$, extending the class of CARMA processes. The definition is based on a system of stochastic partial differential equations which generalize the classical state-space representation of CARMA processes. The resulting CARMA model differs fundamentally from the isotropic CARMA random field of Brockwell and Matsuda. We show existence of the model under mild assumptions and examine some of its features including the second-order structure and path properties. In particular, we investigate the sampling behavior and formulate conditions for the causal CARMA random field to be an ARMA random field when sampled on an equidistant lattice.

中文翻译：

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Lévy 驱动的因果 CARMA 随机场

我们在 $\mathbb{R}^d$ 上引入了 L\'evy 驱动的因果 CARMA 随机场，扩展了 CARMA 过程的类别。该定义基于一个随机偏微分方程系统，它概括了 CARMA 过程的经典状态空间表示。由此产生的 CARMA 模型与 Brockwell 和 Matsuda 的各向同性 CARMA 随机场有着根本的不同。我们在温和的假设下展示了模型的存在，并检查了它的一些特征，包括二阶结构和路径属性。特别是，我们研究了在等距晶格上采样时因果 CARMA 随机场成为 ARMA 随机场的采样行为并制定了条件。