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

2025/10/08 by Vincent Bansaye, Bansaye, Vincent, Bertolino, Alexandre +2
Computer Science · Mathematics · Medicine · #Analysis of PDEs (math.AP) #Bayesian Methods and Mixture Models #FOS: Mathematics #Mathematical and Theoretical Epidemiology and Ecology Models #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2510.07138

openalex publication_date 2025/10/08 · openalex created_date 2025/10/13 · openalex updated_date 2026/07/28

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.

Citations

• The Cauchy Problem For Quasi-Linear Parabolic Systems Revisited
• Perturbative global solutions of a large class of cross diffusion systems in any dimension
• Stability of a cross-diffusion system and approximation by repulsive random walks: a duality approach
• Concentration Inequalities for Ultra Log-Concave Distributions
• Persisting entropy structure for nonlocal cross-diffusion systems
• Weak-strong uniqueness of renormalized solutions to reaction-cross-diffusion systems
• A note on the uniqueness of weak solutions to a class of cross-diffusion systems
• Entropic structure and duality for multiple species cross-diffusion systems
• On the entropic structure of reaction-cross diffusion systems
• New results for triangular reaction cross diffusion system
• Non local Lotka-Volterra system with cross-diffusion in an heterogeneous\n medium

Related

• Stability of a cross-diffusion system and approximation by repulsive random walks: a duality approach
• Convergence towards the population cross-diffusion system from stochastic many-particle system
• Interacting particle approximation of cross-diffusion systems
• Qualitative behavior of solutions to cross-diffusion systems from\n population dynamics
• Beyond Wentzell-Freidlin: semi-deterministic approximations for diffusions with small noise and a repulsive critical boundary point