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 and only if $\alpha\in\mathbb{N}$. 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，线性网络（更一般地，紧致度量图）上的高斯场引起了广泛兴趣。一种在度量图$\Gamma$上定义此类场的策略是通过图度量意义下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$L^{\alpha/2} (\tau u) = \mathcal{W}$，其中对于（充分光滑的）函数$\kappa, a$，有$L = \kappa^2 - \nabla(a\nabla)$，而$\mathcal{W}$是高斯白噪声。我们研究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满足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）为马尔可夫场当且仅当$\alpha\in\mathbb{N}$。进一步地，若$\alpha\in\mathbb{N}$，广义Whittle-Matérn场$u$是$\alpha$阶马尔可夫场，这意味着对于区域$S\subset\Gamma$，给定$u$及其$\alpha-1$阶导数在边界$\partial S$上的值，场$u$在$S$内与$\Gamma\setminus S$中的值条件独立。最后，我们给出基于所发展理论的两个推论：第一，我们证明马尔可夫性质蕴含$u$在固定边$e$上的显式特征刻画，揭示出给定与边$e$相连的两个顶点处的值时，$u$在$e$上的条件分布与$\Gamma$的几何结构无关；第二，我们证明$\Gamma$上方程$L^{1/2}(\tau u) = \mathcal{W}$的解可通过在每条边上施加$\alpha=1$且具有诺伊曼边界条件的独立广义Whittle-Matérn过程，并要求其在顶点处连续来得到。