Continuous-time Markov processes on infinite dimensional hypercube

The infinite-dimensional hypercube (IDH) is the set [Formula: see text] of all finite subsets of [Formula: see text] (the nonnegative integer set) as a graph, where two sets are adjacent if they differ only by precisely one nonnegative integer. In this paper, we investigate the IDH from a perspective of continuous-time Markov processes. First, we introduce a weighted graph Laplacian [Formula: see text] on [Formula: see text] and prove some technical theorems concerning [Formula: see text]. Then we consider the point evaluation process [Formula: see text] on [Formula: see text] and prove that [Formula: see text] is a continuous-time Markov process with [Formula: see text] as its generator. We also construct a class of martingales in terms of [Formula: see text] and [Formula: see text] and obtain some relevant results. Finally, we construct a [Formula: see text]-process [Formula: see text] on [Formula: see text] through a [Formula: see text]-matrix associated with [Formula: see text] and verify its Markov property as well as other properties. Some other related results are also obtained in this paper.