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：近期，线性网络以及更一般的紧度量图上的高斯过程引起了广泛关注。一种在度量图 $\Gamma$ 上定义此类过程的策略是通过在图上的度量下具有各向同性的协方差函数。另一种策略是通过 $\Gamma$ 上的分数阶微分方程 $L^\alpha (\tau u) = \mathcal{W}$，其中 $L = \kappa^2 - \nabla(a\nabla)$，$\kappa, a$ 为足够光滑的函数，而 $\mathcal{W}$ 为高斯白噪声。我们研究了这两类场的马尔可夫性质。我们首先证明，在一般度量图上不存在同时具有各向同性和马尔可夫性的高斯随机场。接着，我们证明第二类场，即广义 Whittle--Matérn 场，是马尔可夫的当且仅当 $\alpha\in\mathbb{N}$；并且当 $\alpha\in\mathbb{N}$ 时，该场是 $\alpha$ 阶马尔可夫的，这实质上意味着在区域 $S\subset\Gamma$ 上的过程在给定 $\partial S$ 上的过程及其 $\alpha-1$ 阶导数值的条件下，与 $\Gamma\setminus S$ 上的过程条件独立。最后，我们证明马尔可夫性质蕴含了在固定边 $e$ 上过程的显式刻画，这特别表明，在给定与 $e$ 相连的两个顶点处的值的条件下，边 $e$ 上过程的条件分布与 $\Gamma$ 的几何形状无关。