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Generalized front propagation for spatial stochastic population models

Thomas Hughes, Jessica Lin

TL;DR

This work develops a unified probabilistic framework to prove that rescaled spatial stochastic population models converge to generalized front propagation governed by mean curvature flow, using moment duality and Barles–Souganidis’ level-set approach. It relaxes regularity assumptions on initial data and yields global-in-time convergence beyond singularities of classical MCF, applying to models studied by Etheridge–Freeman–Penington and Huang–Durrett and related systems. By translating front-propagation conditions into verifiable properties of the dual/ voting algorithm, the paper broadens the applicability of generalized front propagation to a wide class of stochastic spatial models, including BBM-based, SLFV, voter-model perturbations, and stirring-based reproductive systems. The results provide a robust, PDE-free pathway to understanding phase separation and interface dynamics in complex biological systems with potential for further extensions to jump processes and more general flows.

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].

Generalized front propagation for spatial stochastic population models

TL;DR

This work develops a unified probabilistic framework to prove that rescaled spatial stochastic population models converge to generalized front propagation governed by mean curvature flow, using moment duality and Barles–Souganidis’ level-set approach. It relaxes regularity assumptions on initial data and yields global-in-time convergence beyond singularities of classical MCF, applying to models studied by Etheridge–Freeman–Penington and Huang–Durrett and related systems. By translating front-propagation conditions into verifiable properties of the dual/ voting algorithm, the paper broadens the applicability of generalized front propagation to a wide class of stochastic spatial models, including BBM-based, SLFV, voter-model perturbations, and stirring-based reproductive systems. The results provide a robust, PDE-free pathway to understanding phase separation and interface dynamics in complex biological systems with potential for further extensions to jump processes and more general flows.

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].
Paper Structure (31 sections, 24 theorems, 184 equations)

This paper contains 31 sections, 24 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Broad Ideas and Informal Discussion of the Main Results
  3. Further Literature Review and Comparison with Other Works
  4. Outline of the Paper
  5. Preliminaries and Statement of the Main Results
  6. Spatial Stochastic Markov processes and Duals
  7. The $g$-function and the Voting Algorithm
  8. Motion By MCF and the Level-Set Method
  9. Statement of the Main Result
  10. Strategy of the Proof: the Abstract Approach of Barles and Souganidis using the Approximate Dual
  11. A Precise Framework for the Voting Algorithm and the Verification of (J2)-(J3)
  12. The Voting Algorithm on Trees
  13. Voting algorithm on the tree induced by the dual process
  14. The $g$-function
  15. Verification of (J2)-(J3)
  16. ...and 16 more sections

Key Result

Theorem 2.6

Suppose that $(w_t^\epsilon, t\geq 0)_{\epsilon > 0}$ is a family of stochastic spatial models such that the following holds: Then for any $p$ defining $\Gamma_{0}\subseteq \mathbb{R}^{\mathbf{d}}$, as $\epsilon\to 0$, $\mathbb{E}^{\epsilon}_{p} [w^{\epsilon}_t]$ converges locally uniformly to $(a,b)$-generalized MCF started from $\Gamma_{0}$.

Theorems & Definitions (50)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Remark 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 40 more