We study the convergence of a sequence of evolution equations for measures supported on the nodes of a graph. The evolution equations themselves can be interpreted as the forward Kolmogorov equations of Markov jump processes, or equivalently as the equations for the concentrations in a network of linear reactions. The jump rates or reaction rates are divided in two classes; ‘slow’ rates are constant, and ‘fast’ rates are scaled as \(1/\epsilon \), and we prove the convergence in the fast-reaction limit \(\epsilon \rightarrow 0\). We establish a \(\Gamma \)-convergence result for the rate functional in terms of both the concentration at each node and the flux over each edge (the level-2.5 rate function). The limiting system is again described by a functional, and characterises both fast and slow fluxes in the system. This method of proof has three advantages. First, no condition of detailed balance is required. Secondly, the formulation in terms of concentration and flux leads to a short and simple proof of the \(\Gamma \)-convergence; the price to pay is a more involved compactness proof. Finally, the method of proof deals with approximate solutions, for which the functional is not zero but small, without any changes.

我们研究了图节点上支持的度量的一系列演化方程的收敛性。演化方程本身可以解释为马尔可夫跳跃过程的正向 Kolmogorov 方程，或者等效地解释为线性反应网络中的浓度方程。跳跃率或反应率分为两类；“慢”速率是恒定的，“快”速率被缩放为 \(1/\epsilon \)，我们证明了快速反应极限\(\epsilon \rightarrow 0\)的收敛性。我们建立一个\(\Gamma \)- 在每个节点的浓度和每个边缘的通量方面的速率函数的收敛结果（2.5 级速率函数）。限制系统再次由泛函描述，并表征系统中的快速和慢速通量。这种证明方法具有三个优点。首先，不需要详细平衡的条件。其次，关于浓度和通量的公式导致\(\Gamma \)收敛的简短证明；付出的代价是更复杂的紧凑性证明。最后，证明方法处理近似解，其泛函不为零而是很小，没有任何变化。