A constrained Wasserstein gradient flow for aggregation–diffusion dynamics with application to economic adjustment

Abstract We study a class of constrained aggregation–diffusion dynamics formulated as a Wasserstein gradient flow with a linear moment constraint. The model describes the evolution of a probability density on R+ governed by a Fokker–Planck equation with potential, interaction and entropic diffusion terms, while preserving a prescribed first moment. The constraint is enforced through a time-dependent Lagrange multiplier that modifies the drift and keeps the trajectory on an affine feasibility manifold. We show that the resulting equation arises as the gradient flow of a free-energy functional in the 2-Wasserstein space of probability measures subject to a linear constraint. Under standard convexity assumptions on the potentials, the evolution admits global weak solutions, preserves positivity and the moment constraint, satisfies an energy dissipation inequality and converges to a constrained equilibrium. To approximate the dynamics, we introduce a constraint-preserving variational discretization based on a projected Jordan–Kinderlehrer–Otto (JKO) scheme. The method maintains mass conservation, enforces the moment constraint at each time step and inherits the dissipative structure of the continuous flow. Numerical experiments illustrate convergence towards the constrained equilibrium, verify energy decay and examine the influence of interaction and diffusion parameters. Interpreting the multiplier as a shadow price provides an application to redistribution and adjustment dynamics in heterogeneous-agent economic systems.