Systems of interacting diffusions with partial annihilation through membranes

Zhen Qing Chen, Wai Tong Fan

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Abstract

We introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface I . This system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. A related interacting random walk model with discrete state spaces has recently been introduced and studied in Chen and Fan (2014). In this paper, we establish the functional law of large numbers for this new system, thereby extending the hydrodynamic limit in Chen and Fan (2014) to reflected diffusions in domains with mixed-type boundary conditions, which include absorption (harvest of electric charges). We employ a new and direct approach that avoids going through the delicate BBGKY hierarchy.

Original languageEnglish
Pages (from-to)100-146
Number of pages47
JournalAnnals of Probability
Volume45
Issue number1
DOIs
Publication statusPublished - 2017
Externally publishedYes

Keywords

  • Annihilation
  • Coupled partial differential equation
  • Dirichlet form
  • Hydrodynamic limit
  • Interacting diffusion
  • Martingales
  • Nonlinear boundary condition
  • Reflected diffusion

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Chen, Z. Q., & Fan, W. T. (2017). Systems of interacting diffusions with partial annihilation through membranes. Annals of Probability, 45(1), 100-146. https://doi.org/10.1214/15-AOP1047