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

TL;DR

This paper demonstrates that condensing zero-range processes, when scaled appropriately, converge to a diffusion process that progressively reduces dimension and eventually becomes trapped at a vertex, revealing a novel boundary absorption mechanism.

Contribution

It introduces a new dimension-decaying diffusion process as the limit of condensing zero-range processes and develops a method to extend the martingale problem domain.

Findings

The limiting process reduces dimension at boundary faces.

The process is absorbed at vertices of the simplex.

The method for extending the martingale problem domain is broadly applicable.

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 \[ \Sigma = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in S} x_i = 1\}, \] 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 \[ \Sigma_A = \{(x_1,\dots,x_S) : x_i \ge 0,\; \sum_{i\in A} x_i = 1\}, \qquad A \subset S, \] 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.