4.6 Article

KINETIC LIMITS FOR PAIR-INTERACTION DRIVEN MASTER EQUATIONS AND BIOLOGICAL SWARM MODELS

Journal

MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
Volume 23, Issue 7, Pages 1339-1376

Publisher

WORLD SCIENTIFIC PUBL CO PTE LTD
DOI: 10.1142/S0218202513500115

Keywords

Master equation; kinetic equations; binary interactions; propagation of chaos; Kac's master equations; swarms; correlation

Funding

  1. Region Midi-Pyrenees
  2. US National Science Foundation [DMS 0901632]
  3. ANR [ANR-08-BLAN-0333-01]
  4. Swedish Research Council
  5. Agence Nationale de la Recherche (ANR) [ANR-08-BLAN-0333] Funding Source: Agence Nationale de la Recherche (ANR)
  6. Division Of Mathematical Sciences
  7. Direct For Mathematical & Physical Scien [1201354] Funding Source: National Science Foundation

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We consider a class of stochastic processes modeling binary interactions in an N-particle system. Examples of such systems can be found in the modeling of biological swarms. They lead to the definition of a class of master equations that we call pair-interaction driven master equations. In the spatially homogeneous case, we prove a propagation of chaos result for this class of master equations which generalizes Mark Kac's well-known result for the Kac model in kinetic theory. We use this result to study kinetic limits for two biological swarm models. We show that propagation of chaos may be lost at large times and we exhibit an example where the invariant density is not chaotic.

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