Wasserstein convergence rates of increasingly concentrating probability measures

For ℓ: Rd→[0,∞) we consider the sequence of probability measures (μn)n∈N, where μn is determined by a density that is proportional to exp(−nℓ). We allow for infinitely many global minimal points of ℓ, as long as they form a finite union of compact manifolds. In this scenario, we show estimates for the p-Wasserstein convergence of (μn)n∈N to its limit measure. Imposing regularity conditions we obtain a speed of convergence of n−1/(2p) and adding a further technical assumption, we can improve this to a p-independent rate of 1/2 for all orders p∈N of the Wasserstein distance.