The notion of Laplacian of a graph can be generalized to simplicial complexes and hypergraphs, and contains information on the topology of these structures. Even for a graph, the consideration of associated simplicial complexes is interesting to understand its shape. Whereas the Laplacian of a graph has a simple probabilistic interpretation as the generator of a continuous time Markov chain on the graph, things are not so direct when considering simplicial complexes. We define here new Markov chains on simplicial complexes. For a given order~$k$, the state space is the set of $k$-cycles that are chains of $k$-simplexes with null boundary. This new framework is a natural generalization of the canonical Markov chains on graphs. We show that the generator of our Markov chain is the upper Laplacian defined in the context of algebraic topology for discrete structure. We establish several key properties of this new process: in particular, when the number of vertices is finite, the Markov chain is positive recurrent. This result is not trivial, since the cycles can loop over themselves an unbounded number of times. We study the diffusive limits when the simplicial complexes under scrutiny are a sequence of ever refining triangulations of the flat torus. Using the analogy between singular and Hodge homologies, we express this limit as valued in the set of currents. The proof of tightness and the identification of the limiting martingale problem make use of the flat norm and carefully controls of the error terms in the convergence of the generator. Uniqueness of the solution to the martingale problem is left open. An application to hole detection is carried.

翻译：图拉普拉斯算子的概念可以推广到单纯复形和超图上，并且包含这些结构的拓扑信息。即使对于图，考虑相关的单纯复形也有助于理解其形状。尽管图的拉普拉斯算子具有简单的概率解释，即作为图上连续时间马尔可夫链的生成元，但在考虑单纯复形时，情况并非如此直接。我们在此定义单纯复形上的新马尔可夫链。对于给定阶数~$k$，状态空间是$k$环的集合，这些环是边界为零的$k$单形链。这一新框架是图上规范马尔可夫链的自然推广。我们证明，我们的马尔可夫链的生成元是在代数拓扑背景下为离散结构定义的上拉普拉斯算子。我们建立了这一新过程的几个关键性质：特别是，当顶点数有限时，马尔可夫链是正遍历的。这一结果并非平凡，因为环可以无界次数地自环。我们研究了当所考察的单纯复形是平坦环面上不断精化的三角剖分序列时的扩散极限。利用奇异同调与霍奇同调之间的类比，我们将该极限表示为电流集合中的取值。紧致性的证明以及极限鞅问题的识别使用了平坦范数，并对生成元收敛中的误差项进行了精细控制。鞅问题解的唯一性仍悬而未决。我们还将该方法应用于空洞检测。