Large systems of diffusions interacting through their ranks

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We study the limiting behavior of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that under certain assumptions the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The implications of the results for rank-based models of capital distributions in financial markets are also explained.

Original languageEnglish (US)
Pages (from-to)1730-1747
Number of pages18
JournalStochastic Processes and their Applications
Issue number4
StatePublished - Apr 2012
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Modeling and Simulation
  • Applied Mathematics


  • Capital distributions
  • Diffusion processes
  • McKean-Vlasov equation
  • Particle method
  • Porous medium equation
  • Rank-based market models


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