We consider a countable system of interacting (possibly non-Markovian) stochastic differential equations driven by independent Brownian motions and indexed by the vertices of a locally finite graph $G=(V,E)$. The drift of the process at each vertex is influenced by the states of that vertex and its neighbors, and the diffusion coefficient depends on the state of only that vertex. Such processes arise in a variety of applications including statistical physics, neuroscience, engineering and math finance. Under general conditions on the coefficients, we show that if the initial conditions form a second-order Markov random field on $d$-dimensional Euclidean space, then at any positive time, the collection of histories of the processes at different vertices forms a second-order Markov random field on path space. We also establish a bijection between (second-order) Gibbs measures on ${\left({\mathbb{R}}^d\right)}^V$ (with finite second moments) and a set of (second-order) Gibbs measures on path space, corresponding respectively to the initial law and the law of the solution to the stochastic differential equation. As a corollary, we establish a Gibbs uniqueness property that shows that for infinite graphs the joint distribution of the paths is completely determined by the initial condition and the specifications, namely the family of conditional distributions on finite vertex sets given the configuration on the complement. Along the way, we establish approximation and projection results for Markov random fields on locally finite graphs that may be of independent interest.

我们考虑由独立布朗运动驱动并由局部有限图的顶点索引的相互作用（可能是非马尔可夫）随机微分方程的可数系统 $G=(伏,乙)$. 每个顶点的过程漂移受该顶点及其相邻顶点的状态影响，而扩散系数仅取决于该顶点的状态。此类过程出现在各种应用中，包括统计物理学、神经科学、工程和数学金融。在系数的一般条件下，我们表明，如果初始条件形成二阶马尔可夫随机场$d$维欧几里得空间，那么在任何正时刻，不同顶点的过程历史的集合在路径空间上形成一个二阶马尔可夫随机场。我们还建立了（二阶）吉布斯测度之间的双射${\left({\mathbb{电阻}}^d\right)}^{伏}$（具有有限的二阶矩）和一组（二阶）吉布斯测度在路径空间上，分别对应于初始定律和随机微分方程的解法。作为推论，我们建立了一个 Gibbs 唯一性属性，它表明对于无限图，路径的联合分布完全由初始条件和规范决定，即给定补集配置的有限顶点集上的条件分布族。在此过程中，我们在可能具有独立兴趣的局部有限图上建立马尔可夫随机场的近似和投影结果。