Continuous-time Markov processes on infinite dimensional hypercube

Jijun Zhao

doi:10.1142/S0219025725500146

Continuous-time Markov processes on infinite dimensional hypercube

2026-01-01

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DOI：10.1142/S0219025725500146

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The infinite-dimensional hypercube (IDH) is the set Gamma of all finite subsets of & Nopf; (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 Delta(w) on Gamma and prove some technical theorems concerning Delta(w) . Then we consider the point evaluation process X on Gamma and prove that X is a continuous-time Markov process with Delta(w) as its generator. We also construct a class of martingales in terms of X and Delta(w) and obtain some relevant results. Finally, we construct a Q-process Y on Gamma through a Q-matrix associated with Delta(w) and verify its Markov property as well as other properties. Some other related results are also obtained in this paper.

Keyword:

Infinite dimensional hypercube

Laplace operator

continuous-time Markov process

Q-matrix

martingale

Continuous-time Markov processes on infinite dimensional hypercube

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