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Stability of non-conservative cross diffusion model and approximation by stochastic particle systems

Vincent Bansaye, Alexandre Bertolino, Ayman Moussa

TL;DR

This work develops a rigorous bridge from microscopic stochastic population dynamics to non-conservative SKT cross-diffusion models. It extends stability analysis to include birth and death reactions via a duality framework under a Petrovskii-type smallness condition, and establishes strong, pathwise convergence of semi-discrete approximations and stochastic particle systems to the macroscopic limit. The authors provide quantitative error bounds that scale with population size $N$ and grid size $M$, supported by martingale estimates and new large deviations results for births and deaths. The results generalize prior conservative results, yield robust convergence in one-dimensional spatial settings, and offer tools potentially extendable to higher dimensions and complex networks. Overall, the paper delivers a comprehensive, quantitative hydrodynamic-type limit for non-conservative SKT models and contributes independently useful large deviation estimates for structured populations.

Abstract

We study the stability of non-conservative deterministic cross diffusion models and prove that they are approximated by stochastic population models when the populations become locally large. In this model, the individuals of two species move, reproduce and die with rates sensitive to the local densities of the two species. Quantitative estimates are given and convergence is obtained soon as the population per site and the number of sites go to infinity. The proofs rely on the extension of stability estimates via duality approach under a smallness condition and the development of large deviation estimates for structured population models, which are of independent interest. The proofs also involve martingale estimates in H^{-1} and improve the approximation results in the conservative case as well.

Stability of non-conservative cross diffusion model and approximation by stochastic particle systems

TL;DR

This work develops a rigorous bridge from microscopic stochastic population dynamics to non-conservative SKT cross-diffusion models. It extends stability analysis to include birth and death reactions via a duality framework under a Petrovskii-type smallness condition, and establishes strong, pathwise convergence of semi-discrete approximations and stochastic particle systems to the macroscopic limit. The authors provide quantitative error bounds that scale with population size and grid size , supported by martingale estimates and new large deviations results for births and deaths. The results generalize prior conservative results, yield robust convergence in one-dimensional spatial settings, and offer tools potentially extendable to higher dimensions and complex networks. Overall, the paper delivers a comprehensive, quantitative hydrodynamic-type limit for non-conservative SKT models and contributes independently useful large deviation estimates for structured populations.

Abstract

We study the stability of non-conservative deterministic cross diffusion models and prove that they are approximated by stochastic population models when the populations become locally large. In this model, the individuals of two species move, reproduce and die with rates sensitive to the local densities of the two species. Quantitative estimates are given and convergence is obtained soon as the population per site and the number of sites go to infinity. The proofs rely on the extension of stability estimates via duality approach under a smallness condition and the development of large deviation estimates for structured population models, which are of independent interest. The proofs also involve martingale estimates in H^{-1} and improve the approximation results in the conservative case as well.

Paper Structure

This paper contains 20 sections, 33 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Main results
  4. Stability estimate and semi discrete approximation for the SKT system
  5. Convergence of the stochastic particle systems
  6. Stability of the continuous and semi-discrete systems
  7. Duality lemma and proof of Theorem \ref{['th:cont_stab']}
  8. Space $\ell^2(\mathbb{T} \xspace_M)$ and discretization of the continuous solution
  9. Stability of pertubated semi-discrete system and proof of Theorem \ref{['th:disc_stab']}
  10. Convergence of stochastic processes
  11. Construction of the process and first estimates
  12. Martingales inequalities
  13. Large deviations estimates for births and deaths
  14. Quantitative estimates and proof of Theorem \ref{['th:stoc_cv']}
  15. Appendix
  16. ...and 5 more sections

Key Result

Theorem 1

Assume that $\mu_i, R_i$ are globally Lipschitz and $\inf_{\mathbb{R} \xspace_+}\mu_i = \alpha_i >0$ for $i \in \{ 1, 2\}$. Then for any $T>0$ and all non-negative solutions $(u,v), (\overline{u}, \overline{v}) \in L^\infty({Q_T} \xspace )$ of eq:SKT with respective initial conditions $(u_0, v_0)$ a where $c$ and the constant behind $\lesssim$ depend only on the $L^\infty$ norm and Lipschitz const

Theorems & Definitions (59)

  • Definition 1.1
  • Theorem 1
  • Remark
  • Corollary 1.1
  • Theorem 2
  • Corollary 1.2
  • Theorem 3
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • ...and 49 more