Transporting a Dirac mass in a mean field planning problem

TL;DR

This paper investigates a mean field planning problem starting with a Dirac mass, establishing the existence of a unique solution that converges to a self-similar profile over time, using rescaling and Lyapunov functionals.

Contribution

It introduces a novel analysis of mean field planning with initial Dirac mass, demonstrating existence, uniqueness, and self-similar convergence of solutions.

Findings

Unique solution exists for the initial Dirac mass problem.

Solutions converge to a self-similar profile as time approaches zero.

Rescaling and Lyapunov functionals effectively characterize the solution's behavior.

Abstract

We study a mean field planning problem in which the initial density is a Dirac mass. We show that there exists a unique solution which converges to a self-similar profile as time tends to $0$. We proceed by studying a continuous rescaling of the solution, and characterizing its behavior near the initial time through an appropriate Lyapunov functional.