Generalized front propagation for spatial stochastic population models

TL;DR

This paper develops a unified framework to prove that rescaled spatial stochastic population models converge to generalized propagating fronts, extending previous results to more general initial conditions and beyond singularities.

Contribution

It introduces a broad, adaptable approach using the level-set method to establish global-in-time convergence of stochastic models to front propagation, surpassing prior regularity and singularity limitations.

Findings

Proves convergence of stochastic models to generalized fronts globally in time.

Removes regularity restrictions on initial data.

Extends convergence results beyond singularities of mean curvature flow.

Abstract

We present a general framework which can be used to prove that, in an annealed sense, rescaled spatial stochastic population models converge to generalized propagating fronts. Our work is motivated by recent results of Etheridge, Freeman, and Penington [EFP2017] and Huang and Durrett [HD2021], who proved convergence to classical mean curvature flow (MCF) for certain spatial stochastic processes, up until the first time when singularities of MCF form. Our arguments rely on the level-set method and the abstract approach to front propagation introduced by Barles and Souganidis [BS1998]. This approach is amenable to stochastic models equipped with moment duals which satisfy certain general and verifiable properties. Our main results improve the existing results in several ways, first by removing regularity conditions on the initial data, and second by establishing convergence beyond the formation of singularities of MCF. In particular, we obtain a general convergence theorem which holds globally in time. This is then applied to all of the models considered in [EFP2017] and [HD2021].