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From boundary random walks to Feller's Brownian Motions

Liping Li, Zhangjie Wang

TL;DR

The paper develops a rigorous invariance principle linking boundary random walks on $\mathbb{N}$ with Feller's Brownian motions on $[0,\infty)$, by embedding both processes in Skorokhod space and proving tightness plus a martingale-problem uniqueness argument. It introduces $n$-dependent boundary-jump schemes that approximate FBMs with general boundary data $(p_1,p_2,p_3,p_4)$, establishes two main convergence results for the regimes $p_3\neq0$ and $p_2\neq0,p_3=0$, and details the corresponding scaling of the BRW jumping measure. The approach relies on Aldous' tightness criterion and a martingale-problem framework to identify the limit without requiring generator convergence, while carefully handling boundary effects and potential jumps via martingale estimates and boundary-time decompositions. The results unify and extend previous discrete approximations of diffusion processes with boundary interactions (absorption, reflection, stickiness, elasticity) to include boundary-jump behavior, providing concrete BRW constructions that converge to the corresponding FBMs and highlighting regimes where purely jump-boundary behavior remains elusive to this method. Practically, this yields principled discrete models for simulating and analyzing diffusions with rich boundary dynamics in one dimension.

Abstract

We establish an invariance principle connecting boundary random walks on $\mathbb N$ with Feller's Brownian motions on $[0,\infty)$. A Feller's Brownian motion is a Feller process on $[0,\infty)$ whose excursions away from the boundary $0$ coincide with those of a killed Brownian motion, while its behavior at the boundary is characterized by a quadruple $(p_1,p_2,p_3,p_4)$. This class encompasses many classical models, including absorbed, reflected, elastic, and sticky Brownian motions, and further allows boundary jumps from $0$ governed by the measure $p_4$. For any Feller's Brownian motion that is not purely driven by jumps at the boundary, we construct a sequence of boundary random walks whose appropriately rescaled processes converge weakly to the given Feller's Brownian motion.

From boundary random walks to Feller's Brownian Motions

TL;DR

The paper develops a rigorous invariance principle linking boundary random walks on with Feller's Brownian motions on , by embedding both processes in Skorokhod space and proving tightness plus a martingale-problem uniqueness argument. It introduces -dependent boundary-jump schemes that approximate FBMs with general boundary data , establishes two main convergence results for the regimes and , and details the corresponding scaling of the BRW jumping measure. The approach relies on Aldous' tightness criterion and a martingale-problem framework to identify the limit without requiring generator convergence, while carefully handling boundary effects and potential jumps via martingale estimates and boundary-time decompositions. The results unify and extend previous discrete approximations of diffusion processes with boundary interactions (absorption, reflection, stickiness, elasticity) to include boundary-jump behavior, providing concrete BRW constructions that converge to the corresponding FBMs and highlighting regimes where purely jump-boundary behavior remains elusive to this method. Practically, this yields principled discrete models for simulating and analyzing diffusions with rich boundary dynamics in one dimension.

Abstract

We establish an invariance principle connecting boundary random walks on with Feller's Brownian motions on . A Feller's Brownian motion is a Feller process on whose excursions away from the boundary coincide with those of a killed Brownian motion, while its behavior at the boundary is characterized by a quadruple . This class encompasses many classical models, including absorbed, reflected, elastic, and sticky Brownian motions, and further allows boundary jumps from governed by the measure . For any Feller's Brownian motion that is not purely driven by jumps at the boundary, we construct a sequence of boundary random walks whose appropriately rescaled processes converge weakly to the given Feller's Brownian motion.
Paper Structure (20 sections, 12 theorems, 157 equations, 1 figure)

This paper contains 20 sections, 12 theorems, 157 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Boundary Random Walk and Feller's Brownian Motion
  3. Boundary Random Walk
  4. Feller's Brownian motion
  5. From BRWs to FBM: Heuristic investigations
  6. Power-law scaling schemes for approximating FBM by BRWs
  7. The case $p_2=p_3=0$
  8. From BRWs to FBM: Rigorous results
  9. Realization on the Skorohod space
  10. Main result for $p_3\neq 0$
  11. Main result for $p_2\neq 0,p_3=0$
  12. The special case $p_4=0$
  13. Proof of Theorem \ref{['THM31']}: Tightness
  14. Strategy of the proof
  15. Proof of \ref{['eq:42']}
  16. ...and 5 more sections

Key Result

Lemma 2.1

Assume that $\mathbb{P}(X_0=i_0)=1$ for some $i_0\in \mathbb{N}$, and that there exists an integer $j_0$ such that $\mathfrak{p}_j=0$ for all $j>j_0$. Then, for any real-valued function $f$ on $\mathbb{N}$, the process defined by is a square-integrable $(\mathscr G^X_m)$-martingale, where $(P-I)f(i):=\sum_{j\in \mathbb{N}}p_{ij}f(j)-f(i)$ for all $i\in \mathbb{N}$.

Figures (1)

  • Figure 2.1: Jumping behavior of BRW.

Theorems & Definitions (32)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Remark 4.1
  • Theorem 4.2
  • ...and 22 more