2020/07/03 by Shuwen Lou, Lou, Shuwen · 1 citation
Mathematics · #60H30 #60J35 #60J45 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Primary 60J60 #Probability (math.PR) #Secondary 31C25 #Stochastic processes and statistical mechanics #math.PR #msc:31C25 #msc:60H30 #msc:60J35 #msc:60J45 #msc:60J60
paper · pdf · doi:10.48550/arxiv.2007.01933
openalex publication_date 2020/07/03 · arxiv created 2021/10/25 · arxiv updated 2021/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we study the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) introduced in [4] by continuous time random walks on square lattices. The state space of BMVD contains a 2-dimensional component, a 3-dimensional component, and a "darning point" which joins these two components. Such a state space is equipped with the geodesic distance, under which BMVD is a diffusion process. In this paper, we prove that BMVD restricted on a bounded domain containing the darning point is the weak limit of continuous time reversible random walks with exponential holding times. Upon each move, except at the "darning point", these random walks jump to any of its nearest neighbors with equal probability. The behavior of such a random walk at the "darning point" is also given explicitly in this paper.