Discrete approximation of reflected Brownian motions by Markov chains on partitions of domains

TL;DR

This work addresses the problem of discretely approximating reflected Brownian motion on a domain by Markov chains defined on general partitions of the domain. The authors construct a corrected generator on partitions that accounts for inhomogeneity and boundary reflection, and prove sufficient conditions under which the associated chain converges to the Neumann Laplacian and, consequently, to $RBM$ in Skorokhod space. They establish convergence results (for interior and boundary behavior) under domain regularity ($D$ being a $C^{1,\alpha}$-domain) and mesh-size conditions, and show that a natural class of test functions forms a core for the limiting generator in favorable cases. The paper further provides examples (e.g., Voronoi partitions) and discusses implications for spectral approximation and potential extensions. Overall, it delivers a general, robust framework for approximating boundary-reflected diffusions via partition-based Markov dynamics with proven convergence.

Abstract

In this paper, we study discrete approximation of reflected Brownian motions on domains in Euclidean space. Our approximation is given by a sequence of Markov chains on partitions of the domain, where we allow uneven or random partitions. We provide sufficient conditions for the weak convergence of the Markov chains.

Figures (1)

• Figure 1: Figure of a partition of $\mkern 1.5mu\overline{\mkern-1.5muD\mkern-1.5mu}\mkern 1.5mu$. The black dot on each cell denotes the center of gravity of that cell. The white dot denotes the nearest point of $\partial D$ to the center of gravity.

Theorems & Definitions (69)

• Definition 2.1
• Lemma 2.2
• Definition 2.3
• Lemma 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Theorem 2.8
• Proposition 2.9
• Remark 2.10