Kinetic Mean-Field Planning via Schrödinger Bridges: Existence, Penalization, and Consistency Bounds

Abstract

We study a mean-field planning problem for a swarm of interacting, non-cooperative agents whose dynamics are stochastic and kinetic. The noise acts on the velocity only, and the interaction (of Cucker-Smale type for instance) enters through the drift. For a fixed flow, the problem reduces to a Schrödinger bridge problem, and we first prove the existence of an equilibrium by a Schauder fixed-point argument. Then we replace the hard terminal constraint by a penalty function and show that equilibria of the penalized problems converge, along subsequences, to an equilibrium of the original problem. Since equilibria are not unique in general, we instead quantify the consistency error, which measures how far an equilibrium of the penalized problem is from being an equilibrium of the original one. The rate is of order $n^{-1/2}$ for the relative-entropy penalty. The proof relies on the quantitative stability of entropic optimal transport and on a pointwise linear growth bound on the score of the degenerate reference dynamics, which is of independent interest.