Global weak martingale solutions to a stochastic two-sidedly degenerate aggregation-diffusion equation issued from biology
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866916990544773120 |
|---|---|
| author | Bendahmane, Mostafa Mehdaoui, Mohamed Tilioua, Mouhcine |
| author_facet | Bendahmane, Mostafa Mehdaoui, Mohamed Tilioua, Mouhcine |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_03947 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global weak martingale solutions to a stochastic two-sidedly degenerate aggregation-diffusion equation issued from biology Bendahmane, Mostafa Mehdaoui, Mohamed Tilioua, Mouhcine Probability 60H15, 35K57, 35A05, 35R60, 92D25 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. |
| title | Global weak martingale solutions to a stochastic two-sidedly degenerate aggregation-diffusion equation issued from biology |
| topic | Probability 60H15, 35K57, 35A05, 35R60, 92D25 |
| url | https://arxiv.org/abs/2510.03947 |