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Global weak martingale solutions to a stochastic two-sidedly degenerate aggregation-diffusion equation issued from biology

Mostafa Bendahmane, Mohamed Mehdaoui, Mouhcine Tilioua

TL;DR

The paper addresses the well-posedness of martingale (probabilistic weak) solutions for a stochastic degenerate aggregation–diffusion equation that models biological clustering with density-dependent diffusion, nonlocal attraction, a reaction term, and environmental noise. It regularizes the degenerate diffusion with a nondegenerate surrogate, constructs Faedo–Galerkin approximations, and uses Prokhorov's tightness and Skorokhod's representation to pass to the limit, finally establishing existence and a duality-based uniqueness result. The study provides a rigorous stochastic framework on bounded domains and demonstrates, via numerical simulations, how environmental noise reshapes aggregation dynamics and long-term behavior, suggesting implications for biological and public-health modeling. The results advance the mathematical foundations for stochastic aggregation–diffusion systems and open directions toward more complex noise structures (e.g., Lévy noise) and long-term stability.

Abstract

The purpose of this paper is to establish the well-posedness of martingale (probabilistic weak) solutions to stochastic degenerate aggregation--diffusion equations arising in biological and public health contexts. The studied equation is of a stochastic degenerate parabolic type, featuring a nonlinear two-sidedly degenerate diffusion term accounting for repulsion, a locally Lipschitz reaction term representing competitive interactions, and a stochastic perturbation term capturing environmental noise and uncertainty in biological systems. The existence of martingale solutions is proved via an auxiliary nondegenerate stochastic system combined with the Faedo--Galerkin method. Convergence of approximate solutions is established through Prokhorov's compactness and Skorokhod's representation theorems, and uniqueness is obtained using a duality approach. Finally, numerical simulations are given to illustrate the impact of environmental noise on aggregation dynamics and the long-term behavior of the system, offering insights that may inspire medical innovation and predictive modeling in public health.

Global weak martingale solutions to a stochastic two-sidedly degenerate aggregation-diffusion equation issued from biology

TL;DR

The paper addresses the well-posedness of martingale (probabilistic weak) solutions for a stochastic degenerate aggregation–diffusion equation that models biological clustering with density-dependent diffusion, nonlocal attraction, a reaction term, and environmental noise. It regularizes the degenerate diffusion with a nondegenerate surrogate, constructs Faedo–Galerkin approximations, and uses Prokhorov's tightness and Skorokhod's representation to pass to the limit, finally establishing existence and a duality-based uniqueness result. The study provides a rigorous stochastic framework on bounded domains and demonstrates, via numerical simulations, how environmental noise reshapes aggregation dynamics and long-term behavior, suggesting implications for biological and public-health modeling. The results advance the mathematical foundations for stochastic aggregation–diffusion systems and open directions toward more complex noise structures (e.g., Lévy noise) and long-term stability.

Abstract

The purpose of this paper is to establish the well-posedness of martingale (probabilistic weak) solutions to stochastic degenerate aggregation--diffusion equations arising in biological and public health contexts. The studied equation is of a stochastic degenerate parabolic type, featuring a nonlinear two-sidedly degenerate diffusion term accounting for repulsion, a locally Lipschitz reaction term representing competitive interactions, and a stochastic perturbation term capturing environmental noise and uncertainty in biological systems. The existence of martingale solutions is proved via an auxiliary nondegenerate stochastic system combined with the Faedo--Galerkin method. Convergence of approximate solutions is established through Prokhorov's compactness and Skorokhod's representation theorems, and uniqueness is obtained using a duality approach. Finally, numerical simulations are given to illustrate the impact of environmental noise on aggregation dynamics and the long-term behavior of the system, offering insights that may inspire medical innovation and predictive modeling in public health.

Paper Structure

This paper contains 20 sections, 13 theorems, 164 equations, 7 figures.

Table of Contents

  1. Introduction and motivation
  2. The general case of a nonlinear density-dependent diffusion coefficient $a : \mathbb{R} \longrightarrow \mathbb{R}^+$
  3. The case of aggregation-diffusion equations in stochastic environments
  4. Preliminaries, notion of weak martingale solutions and main result
  5. Construction of a sequence of Faedo-Galerkin solutions to a non-degenerate system
  6. The non-degenerate system
  7. A sequence of Faedo-Galerkin solutions
  8. Almost-sure positiveness and boundedness of the sequence of Faedo-Galerkin solutions
  9. Uniform a priori and temporal translation estimates
  10. Uniform a priori estimates
  11. Retrieval of unique weak martingale solutions in the degenerate case
  12. Tightness and Skorokhod almost-sure representations
  13. Existence
  14. Uniqueness
  15. A numerical illustration
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let Assumptions a1-a5 be satisfied. Assume further that $0 \leq u_0 \leq \overline{u},\; \mathbb{P}\text{-a.s},$ such that the (initial) probability measure $\mu_{u_0}$ on $L^2(\mathcal{O})$ satisfies the following moment conditition and $\sigma(0)=\sigma(\overline{u})=0$. Then, it holds that Model basicstoch has a weak almost-surely positive martingale solution. Moreover, if $(u_1,\mathcal{W})$

Figures (7)

  • Figure 1: Evolution of the density $u$ at four points in time: $t=0$, $t=4$, $t=8$ and $t=12$ with $\alpha=0.4$ and $\mu=0.5$ in the absence of stochastic noise $(\sigma \equiv 0)$.
  • Figure 2: Evolution of the density $u$ at four points in time: $t=0$, $t=4$, $t=8$ and $t=12$, with $\alpha=0.4$ and $\mu=0.5$ in the presence of the stochastic noise $\sigma(u)=1.2 \min\{u,\overline{u}-u\}$.
  • Figure 3: Evolution of the density $u$ at four points in time: $t=0$, $t=4$, $t=8$ and $t=12$, with $\alpha=0.4$ and $\mu=0.5$ in the presence of the stochastic noise $\sigma(u)=1.2 \sin(\dfrac{\pi u}{\overline{u}})$.
  • Figure 4: Evolution of the density $u$ at four points in time: $t=0$, $t=4$, $t=8$ and $t=12$, with $\alpha=0.4$ and $\mu=0.5$ in the presence of the stochastic noise $\sigma(u)=1.2 \min\{u,\overline{u}-u\}$ for a stochastically-perturbed initial condition $u_0$.
  • Figure 5: Evolution of the density $u$ at four points in time: $t=0$, $t=4$, $t=8$ and $t=12$, with $\alpha=0.4$ and $\mu=0.5$ in the presence of the stochastic noise $\sigma(u)=1.2 \sin(\dfrac{\pi u}{\overline{u}})$ for a stochastically-perturbed initial condition $u_0$.
  • ...and 2 more figures

Theorems & Definitions (33)

  • Remark 2.1
  • Definition 2.1
  • Remark 2.2
  • Theorem 2.1
  • Remark 3.1
  • Remark 3.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 23 more