2021/10/25 by Shuwen Lou, Lou, Shuwen · 2 citations
Economics, Econometrics and Finance · Mathematics · #60J35 #60J65 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Primary 60J27 #Probability (math.PR) #Secondary 31C25 #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.PR #msc:31C25 #msc:60J27 #msc:60J35 #msc:60J65
paper · pdf · doi:10.48550/arxiv.2110.12716
arXiv admin note: text overlap with arXiv:2007.01933
openalex publication_date 2021/10/25 · arxiv created 2021/11/14 · arxiv updated 2021/11/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish the discrete approximation to Brownian motion with varying dimension (BMVD in abbreviation) by random walks. The setting is very similar to that in [11], but here we use a different method allowing us to get rid the restrictions in [11] (or [3]) that the underlying state space has to be bounded, and that the initial distribution of the limiting continuous process has to be its invariant distribution. The approach in this paper is that we first obtain heat kernel upper bounds for the approximating random walks that are uniform in their mesh size, by establishing a Nash-type inequality based on their Dirichlet form characterization. Using the heat kernel upper bound, we then show the tightness of the approximating random walks by delicate analysis.