There has recently been much interest in Gaussian fields on linear networks and, more generally, on compact metric graphs. One proposed strategy for defining such fields 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 $L^{\alpha/2} (\tau u) = \mathcal{W}$ 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. First, we show that no Gaussian random fields exist on general metric graphs that are both isotropic and Markov. Then, we show that the second type of fields, the generalized Whittle--Mat\'ern fields, are Markov if $\alpha\in\mathbb{N}$, and conversely, if $a$ and $\kappa$ are constant and $u$ is Markov, then $\alpha\in\mathbb{N}$. Further, if $\alpha\in\mathbb{N}$, a generalized Whittle--Mat\'ern field $u$ is Markov of order $\alpha$, which means that the field $u$ in one region $S\subset\Gamma$ is conditionally independent of $u$ in $\Gamma\setminus S$ given the values of $u$ and its $\alpha-1$ derivatives on $\partial S$. Finally, we provide two results as consequences of the theory developed: first we prove that the Markov property implies an explicit characterization of $u$ on a fixed edge $e$, revealing that the conditional distribution of $u$ on $e$ given the values at the two vertices connected to $e$ is independent of the geometry of $\Gamma$; second, we show that the solution to $L^{1/2}(\tau u) = \mathcal{W}$ on $\Gamma$ can obtained by conditioning independent generalized Whittle--Mat\'ern processes on the edges, with $\alpha=1$ and Neumann boundary conditions, on being continuous at the vertices.

翻译：近年来，人们对线性网络乃至更一般的紧致度量图上的高斯场产生了浓厚兴趣。在度量图 $\Gamma$ 上定义此类场的一种策略是通过在图上的度量中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 $L^{\alpha/2} (\tau u) = \mathcal{W}$，其中 $L = \kappa^2 - \nabla(a\nabla)$（对于足够好的函数 $\kappa, a$），且 $\mathcal{W}$ 是高斯白噪声。我们研究了这两类场的马尔可夫性质。首先，我们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。其次，我们证明第二类场，即广义 Whittle–Matérn 场，当 $\alpha\in\mathbb{N}$ 时是马尔可夫的；反之，如果 $a$ 和 $\kappa$ 是常数且 $u$ 是马尔可夫的，则 $\alpha\in\mathbb{N}$。此外，若 $\alpha\in\mathbb{N}$，广义 Whittle–Matérn 场 $u$ 是 $\alpha$ 阶马尔可夫的，这意味着区域 $S\subset\Gamma$ 内的场 $u$ 在给定边界 $\partial S$ 上 $u$ 及其 $\alpha-1$ 阶导数值的条件下，与 $\Gamma\setminus S$ 中的 $u$ 条件独立。最后，我们基于所发展的理论给出了两个结果：首先，我们证明马尔可夫性意味着在固定边 $e$ 上 $u$ 的一个显式刻画，揭示了给定连接到 $e$ 的两个顶点处的值后，$e$ 上 $u$ 的条件分布与 $\Gamma$ 的几何形状无关；其次，我们表明 $\Gamma$ 上方程 $L^{1/2}(\tau u) = \mathcal{W}$ 的解可以通过对边上的独立广义 Whittle–Matérn 过程（取 $\alpha=1$ 并满足 Neumann 边界条件）在顶点处连续的条件化而获得。