Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation

Abstract

We study a quasilinear parabolic Cauchy problem with a cumulative distribution function on the real line as an initial condition. We call 'probabilistic solution' a weak solution which remains a cumulative distribution function at all times. We prove the uniqueness of such a solution and we deduce the existence from a propagation of chaos result on a system of scalar diffusion processes, the interactions of which only depend on their ranking. We then investigate the long time behaviour of the solution. Using a probabilistic argument and under weak assumptions, we show that the flow of the Wasserstein distance between two solutions is contractive. Under more stringent conditions ensuring the regularity of the probabilistic solutions, we finally derive an explicit formula for the time derivative of the flow and we deduce the convergence of solutions to equilibrium.

Stochastic partial differential equations: analysis and computations (2013) http://dx.doi.org/10.1007/s40072-013-0014-2

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• Atlas models of equity markets
• Uniform convergence to equilibrium for granular media
• Propagation of chaos and Poincaré inequalities for a system of particles interacting through their cdf
• Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme
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Cited by in corpus (4)

• Approximation of Optimal Transport problems with marginal moments constraints
• The small noise limit of order-based diffusion processes
• Equilibrium large deviations for mean-field systems with translation invariance
• Large Rank-Based Models with Common Noise