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Dimension-decaying diffusion processes as the scaling limit of condensing zero-range processes

Johel Beltrán, Kyuhyeon Choi, Claudio Landim

TL;DR

The article analyzes the diffusive scaling limit of condensing zero-range processes on a finite set, proving convergence to a diffusion on the simplex $\Sigma$ that is absorbed at the boundary. The diffusion exhibits a dimension-decaying mechanism: upon hitting a face $\Sigma_A$, it remains there and evolves in the corresponding lower-dimensional simplex, with parameters depending on $A$, repeating until absorption at a vertex. A central methodological advance is an extension of the martingale problem's domain, enabling functions with boundary discontinuities and facilitating the construction of superharmonic functions essential for proving absorption. The authors establish existence and uniqueness of the limiting diffusion by combining a tightness/convergence argument with a robust absorbing-property analysis, providing a general framework that may apply to other metastable-diffusion contexts and boundary-domain extensions.

Abstract

In this article, we prove that, on the diffusive time scale, condensing zero-range processes converge to a dimension-decaying diffusion process on the simplex \[ Σ= \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in S} x_i = 1\}, \] where $S$ is a finite set. This limiting diffusion has the distinctive feature of being absorbed at the boundary of the simplex. More precisely, once the process reaches a face \[ Σ_A = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in A} x_i = 1\}, \qquad A \subset S, \] it remains confined to this set and evolves in the corresponding lower-dimensional simplex according to a new diffusion whose parameters depend on the subset $A$. This mechanism repeats itself, leading to successive reductions of the dimension, until one of the vertices of the simplex is reached in finite time. At that point, the process becomes permanently trapped. The proof relies on a method to extend the domain of the associated martingale problem, which may be of independent interest and useful in other contexts.

Dimension-decaying diffusion processes as the scaling limit of condensing zero-range processes

TL;DR

The article analyzes the diffusive scaling limit of condensing zero-range processes on a finite set, proving convergence to a diffusion on the simplex that is absorbed at the boundary. The diffusion exhibits a dimension-decaying mechanism: upon hitting a face , it remains there and evolves in the corresponding lower-dimensional simplex, with parameters depending on , repeating until absorption at a vertex. A central methodological advance is an extension of the martingale problem's domain, enabling functions with boundary discontinuities and facilitating the construction of superharmonic functions essential for proving absorption. The authors establish existence and uniqueness of the limiting diffusion by combining a tightness/convergence argument with a robust absorbing-property analysis, providing a general framework that may apply to other metastable-diffusion contexts and boundary-domain extensions.

Abstract

In this article, we prove that, on the diffusive time scale, condensing zero-range processes converge to a dimension-decaying diffusion process on the simplex where is a finite set. This limiting diffusion has the distinctive feature of being absorbed at the boundary of the simplex. More precisely, once the process reaches a face it remains confined to this set and evolves in the corresponding lower-dimensional simplex according to a new diffusion whose parameters depend on the subset . This mechanism repeats itself, leading to successive reductions of the dimension, until one of the vertices of the simplex is reached in finite time. At that point, the process becomes permanently trapped. The proof relies on a method to extend the domain of the associated martingale problem, which may be of independent interest and useful in other contexts.
Paper Structure (16 sections, 43 theorems, 289 equations)

This paper contains 16 sections, 43 theorems, 289 equations.

Table of Contents

  1. Introduction
  2. Model and Main Results
  3. The trace process
  4. The projection map
  5. Extension of the martingale problem
  6. Extension map
  7. A norm map and related Lemmas
  8. Proof of Theorem \ref{['thm:emp']}
  9. A $\mu$-topology on ${\mathcal{E}}_S$
  10. Proof of Proposition \ref{['prop:mudense']}
  11. Proof of Theorem \ref{['thm:abs']}
  12. A superharmonic function
  13. First time interval
  14. Absorption at the boundary
  15. Proof of Theorem \ref{['thm:mainthm']}
  16. ...and 1 more sections

Key Result

Theorem 2.4

For each $x\in \Sigma$, there exists a unique probability measure on $C({\mathbb R}_+, \Sigma)$, denoted by ${\mathbb P}_x$, which starts at $x$ and is a solution of the $({\mathfrak L},{\mathcal{D}}_S)$-martingale problem. Furthermore, let ${\mathbb P}^N_{x_N}$ be the probability measure on $D({\ma

Theorems & Definitions (86)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Proposition 2.8
  • Remark 2.9
  • Lemma 3.1
  • ...and 76 more