Asymptotics of discrete Schrödinger bridges via chaos decomposition

Consider the problem of matching two independent i.i.d. samples of size N from two distributions P and Q in Rd. For an arbitrary continuous cost function, the optimal assignment problem looks for the matching that minimizes the total cost. We consider instead in this paper the problem where each matching is endowed with a Gibbs probability weight proportional to the exponential of the negative total cost of that matching. Viewing each matching as a joint distribution with N atoms, we then take a convex combination with respect to the above Gibbs probability measure. We show that this resulting random joint distribution converges, as N→∞, to the solution of a variational problem, introduced by Föllmer, called the Schrödinger problem. We also prove a limiting Gaussian fluctuation for this convergence in the form of central limit theorems for integrated test functions. This establishes a novel passage for the transition from discrete to continuum in Schrödinger's lazy gas experiment.