Markov Chains on Graphs and Brownian Motion

AbstractWe consider random walks with small fixed steps inside of edges of a graph $${\mathcal{G}}$$ , prescribing a natural rule of probabilities of jumps over a vertex. We show that after an appropriate rescaling such random walks weakly converge to the natural Brownian motion on $${\mathcal{G}}$$ constructed in Ref. 1.