Large systems of diffusions interacting through their ranks

We study the limiting behaviour of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that the limiting dynamics is given by a McKean-Vlasov evolution equation. Moreover, we show that in a wide range of cases the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The uniqueness theory for the latter is used to establish the uniqueness of solutions of the limiting McKean-Vlasov equation and the law of large numbers for the corresponding systems of interacting diffusions. The implications of the results for rank-based models of capital distributions in financial markets are also explained.