There has recently been much interest in Gaussian processes on linear networks and more generally on compact metric graphs. One proposed strategy for defining such processes on a metric graph $\Gamma$ is through a covariance function that is isotropic in a metric on the graph. Another is through a fractional order differential equation $L^\alpha (\tau u) = \mathcal{W}$ on $\Gamma$, where $L = \kappa^2 - \nabla(a\nabla)$ for (sufficiently nice) functions $\kappa, a$, and $\mathcal{W}$ is Gaussian white noise. We study Markov properties of these two types of fields. We first show that there are no Gaussian random fields on general metric graphs that are both isotropic and Markov. We then show that the second type of fields, the generalized Whittle--Mat\'ern fields, are Markov if and only if $\alpha\in\mathbb{N}$, and if $\alpha\in\mathbb{N}$, the field is Markov of order $\alpha$, which essentially means that the process in one region $S\subset\Gamma$ is conditionally independent the process in $\Gamma\setminus S$ given the values of the process and its $\alpha-1$ derivatives on $\partial S$. Finally, we show that the Markov property implies an explicit characterization of the process on a fixed edge $e$, which in particular shows that the conditional distribution of the process on $e$ given the values at the two vertices connected to $e$ is independent of the geometry of $\Gamma$.

翻译：紧致度量图上高斯随机场的马尔科夫特性 转化后摘要： 最近，人们对线性网络和更一般的紧致度量图上的高斯过程产生了极大的兴趣。定义度量图 $\Gamma$ 上的这种过程的一种策略是通过具有图上度量各向同性的协方差函数。另一种策略是通过度量图 $\Gamma$ 上的分数阶微分方程 $L^\alpha (\tau u) = \mathcal{W}$ 来定义它们，其中 $L = \kappa^2 - \nabla(a\nabla)$ 是（足够漂亮的）函数 $\kappa, a$ 和 $\mathcal{W}$ 是高斯白噪声。我们研究了这两种类型的场的马尔科夫特性。首先，我们证明一般度量图上没有同时具有各向同性和马尔科夫性的高斯随机场。然后，我们表明第二种的场——广义的Whittle--Mat\'ern场在 $\alpha \in \mathbb{N}$ 的情况下，当且仅当 $\alpha \in \mathbb{N}$ 时是马尔科夫的，并且如果 $\alpha\in\mathbb{N}$，则场是 $\alpha$ 阶马尔科夫的, 这基本上意味着给定该过程及其 $\alpha-1$ 个导数在 $\partial S$ 上的值，在一个区域 $S\subset\Gamma$ 中的过程在 $\Gamma\setminus S$ 中的过程是条件独立的。最后，我们表明，马尔科夫特性暗示了在一个固定的边缘 $e$ 上的过程的显式特征，这特别显示了给定连接到 $e$ 的两个顶点的值时，$e$ 上的过程的条件分布并不依赖于 $\Gamma$ 的几何形状。