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Global martingale solutions for stochastic Shigesada-Kawasaki-Teramoto population models

Marcel Braukhoff, Florian Huber, Ansgar Jüngel

TL;DR

The paper addresses the challenge of global existence for stochastic SKT cross-diffusion systems with multiplicative noise by exploiting an entropy structure and a novel entropy-regularization scheme. A regularization operator $R_ obreak ext{e}_ obreak s$ built from a connection operator $L$ yields a tractable approximate problem whose entropy inequality provides the needed gradient bounds, higher-order moments, and fractional time regularity. Using tightness in sub-Polish spaces and the Skorokhod–Jakubowski framework, the authors obtain global nonnegative martingale solutions for the case with self-diffusion in any dimension and, in two dimensions, also for the case without self-diffusion under a growth condition on the noise. The results rely on detailed-balance diffusion, entropy-dissipation, and carefully controlled stochastic terms, with examples illustrating admissible colored noises and Lipschitz multiplicative diffusion that preserve the gradient-flow structure. These findings extend the deterministic entropy methods to stochastic SKT systems and provide robust existence results relevant for stochastic population segregation models.

Abstract

The existence of global nonnegative martingale solutions to cross-diffusion systems of Shigesada-Kawasaki-Teramoto type with multiplicative noise is proven. The model describes the stochastic segregation dynamics of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. The diffusion matrix is generally neither symmetric nor positive semidefinite, which excludes standard methods for evolution equations. Instead, the existence proof is based on the entropy structure of the model, a novel regularization of the entropy variable, higher-order moment estimates, and fractional time regularity. The regularization technique is generic and is applied to the population system with self-diffusion in any space dimension and without self-diffusion in two space dimensions.

Global martingale solutions for stochastic Shigesada-Kawasaki-Teramoto population models

TL;DR

The paper addresses the challenge of global existence for stochastic SKT cross-diffusion systems with multiplicative noise by exploiting an entropy structure and a novel entropy-regularization scheme. A regularization operator built from a connection operator yields a tractable approximate problem whose entropy inequality provides the needed gradient bounds, higher-order moments, and fractional time regularity. Using tightness in sub-Polish spaces and the Skorokhod–Jakubowski framework, the authors obtain global nonnegative martingale solutions for the case with self-diffusion in any dimension and, in two dimensions, also for the case without self-diffusion under a growth condition on the noise. The results rely on detailed-balance diffusion, entropy-dissipation, and carefully controlled stochastic terms, with examples illustrating admissible colored noises and Lipschitz multiplicative diffusion that preserve the gradient-flow structure. These findings extend the deterministic entropy methods to stochastic SKT systems and provide robust existence results relevant for stochastic population segregation models.

Abstract

The existence of global nonnegative martingale solutions to cross-diffusion systems of Shigesada-Kawasaki-Teramoto type with multiplicative noise is proven. The model describes the stochastic segregation dynamics of an arbitrary number of population species in a bounded domain with no-flux boundary conditions. The diffusion matrix is generally neither symmetric nor positive semidefinite, which excludes standard methods for evolution equations. Instead, the existence proof is based on the entropy structure of the model, a novel regularization of the entropy variable, higher-order moment estimates, and fractional time regularity. The regularization technique is generic and is applied to the population system with self-diffusion in any space dimension and without self-diffusion in two space dimensions.
Paper Structure (23 sections, 32 theorems, 195 equations)

This paper contains 23 sections, 32 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Notation and main result
  3. Notation and stochastic framework
  4. Assumptions
  5. Main results
  6. Operator setup
  7. Definition of the connection operator $L$
  8. Definition of the regularization operator $R_\varepsilon$
  9. Existence of approximate solutions
  10. Proof of Theorem \ref{['thm.skt']}
  11. Uniform estimates
  12. Tightness of the laws of $(u^\varepsilon)$
  13. Convergence of $(u^\varepsilon)$
  14. Proof of Theorem \ref{['thm.skt2']}
  15. Uniform estimates
  16. ...and 8 more sections

Key Result

Theorem 1

Let $a_{ij}\ge 0$ satisfy the detailed-balance condition, let the stochastic diffusion $\sigma_{ij}$ be Lipschitz continuous on the space of Hilbert--Schmidt operators, and let a certain interaction condition between the entropy and stochastic diffusion hold (see Assumption (A5) below). Then there e

Theorems & Definitions (61)

  • Theorem 1: Informal statement
  • Remark 2: Discussion of the assumptions
  • Remark 3: Reaction terms
  • Definition 1: Martingale solution
  • Theorem 4: Existence for the SKT model with self-diffusion
  • Theorem 5: Existence for the SKT model without self-diffusion
  • Remark 6: Nonnegativity of the solution
  • Proposition 7: Operator $L$
  • proof
  • Lemma 8: Operator $L^{-1}$
  • ...and 51 more