Large Deviations for Diffusions Interacting Through Their Ranks |
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| Authors: | Amir Dembo Mykhaylo Shkolnikov S R Srinivasa Varadhan Ofer Zeitouni |
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| Institution: | 1. Departments of Statistics and Mathematics, Stanford University, Stanford, CA, USA;2. Department of Operations Research and Financial Engineering, Princeton University, Princeton, NJ, USA;3. Courant Institute, New York, NY, USA;4. Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel |
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| Abstract: | We prove a large deviations principle (LDP) for systems of diffusions (particles) interacting through their ranks when the number of particles tends to infinity. We show that the limiting particle density is given by the unique solution of the appropriate McKean‐Vlasov equation and that the corresponding cumulative distribution function evolves according to a nondegenerate generalized porous medium equation with convection. The large deviations rate function is provided in explicit form. This is the first instance of an LDP for interacting diffusions where the interaction occurs both through the drift and the diffusion coefficients and where the rate function can be given explicitly. In the course of the proof, we obtain new regularity results for tilted versions of such a generalized porous medium equation.© 2016 Wiley Periodicals, Inc. |
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