  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.
