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Scaling limit of boundary random walks: A martingale problem approach

Juan Carlos Arroyave, Eldon Barros, Eduardo Pimenta

TL;DR

This work studies the scaling limit of $(\alpha,\beta,A,B)$-boundary random walks on the half-line and shows convergence to Brownian-type diffusions with rich boundary behavior. It adopts a fully probabilistic martingale-problem framework and leverages detailed asymptotics of the discrete local time at the boundary to identify the continuum limit, proving convergence in the $J_1$-Skorokhod topology. The main contributions include a complete phase diagram linking the discrete boundary rates to Knight's coefficients $(c_1,c_2,c_3)$ and a rigorous identification of multiple limiting diffusions: mixed, sticky, elastic, exponential-holding, absorbed, reflected, and killed Brownian motions. The results establish well-posedness of the corresponding martingale problems and provide a robust discrete-to-continuum construction of half-line diffusions, with implications for models incorporating boundary interactions via local time.

Abstract

We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under appropriate scaling, the process converges to the general Brownian motion in the $J_1$-Skorokhod topology. The main novelty of our approach lies in a result on the asymptotic behavior of the local time of the boundary random walks, allowing us to derive a CLT result for several Brownian-type limit processes on the half-line.

Scaling limit of boundary random walks: A martingale problem approach

TL;DR

This work studies the scaling limit of -boundary random walks on the half-line and shows convergence to Brownian-type diffusions with rich boundary behavior. It adopts a fully probabilistic martingale-problem framework and leverages detailed asymptotics of the discrete local time at the boundary to identify the continuum limit, proving convergence in the -Skorokhod topology. The main contributions include a complete phase diagram linking the discrete boundary rates to Knight's coefficients and a rigorous identification of multiple limiting diffusions: mixed, sticky, elastic, exponential-holding, absorbed, reflected, and killed Brownian motions. The results establish well-posedness of the corresponding martingale problems and provide a robust discrete-to-continuum construction of half-line diffusions, with implications for models incorporating boundary interactions via local time.

Abstract

We establish the scaling limit of a class of boundary random walks to the full spectrum of Brownian-type processes on the half-line. By solving the associated martingale problem and employing weak convergence techniques, we prove that under appropriate scaling, the process converges to the general Brownian motion in the -Skorokhod topology. The main novelty of our approach lies in a result on the asymptotic behavior of the local time of the boundary random walks, allowing us to derive a CLT result for several Brownian-type limit processes on the half-line.

Paper Structure

This paper contains 25 sections, 15 theorems, 92 equations, 1 figure.

Table of Contents

  1. introduction
  2. The framework
  3. Notation
  4. Discrete model
  5. Limiting Diffusions: The general Brownian motion
  6. Main result
  7. Local time of boundary random walks at origin
  8. Asymptotic Analysis of Return Probabilities and Expected Local Time
  9. Tightness of the $(\alpha,\beta,A,B)$-boundary random walks
  10. Characterization of the limit points
  11. The mixed Brownian motion as a scaling limit of the $(2,1,A,B)$-boundary random walk
  12. Characterization of limit points
  13. The Sticky Brownian motion as a scaling limit of the $(\alpha,1,A,B)$-random walk with parameters $\alpha > 2$
  14. Characterization of limit points
  15. The exponential holding Brownian motion as a scaling limit of the $(2,\beta,A,B)$-random walk with parameters $\beta > 2$
  16. ...and 10 more sections

Key Result

Theorem 2.3

Knight1981 Any general Brownian motion ${\mathsf W}$ on $[0,\infty)$ has generator ${\mathsf L} = \frac{1}{2}\frac{d^2}{dx^2}$ with corresponding domain for some $c_i \geq 0$ such that $c_1 + c_2 + c_3 = 1$ and $c_1\neq 1$.

Figures (1)

  • Figure 1: Jump rates for the boundary random walk.

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4: Scaling limit to the general Brownian motion
  • Theorem 3.1
  • Corollary 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 19 more