A gradient flow perspective on McKean-Vlasov equations in econophysics

Abstract: We prove that the Gini coefficient of economic inequality is a Lyapunov functional for a class of nonlinear, nonlocal integro-differential equations arising at the intersection of mathematics, economics, and statistical physics. Next, a novel Riemannian geometry is imposed on a subset of probability densities such that the evolutionary dynamics are formally driven by the Gini coefficient functional as a gradient flow. Thus in the same way that classical 2-Wasserstein theory connects heat flow and the Second Law of Thermodynamics by way of Boltzmann entropy, the work here gives rise to a principle of econophysics that is much of the same flavor but for the Gini coefficient. The noncanonical Onsager operators associated to the metric tensors are derived and some transport inequalities proven. The new metric relates to the dual norm of a second-order Sobolev-like factor space, in a similar way to how the classical 2-Wasserstein metric linearizes as the dual norm of a first-order, homogeneous Sobolev space.