Stochastic sewing lemma on Wasserstein space

The stochastic sewing lemma recently introduced by Lê [20] allows to construct a unique limit process from a doubly indexed stochastic process that satisfies some regularity. This lemma is stated in a given probability space on which these processes are defined. The present paper develops a version of this lemma for probability measures: from a doubly indexed family of maps on the set of probability measures that have a suitable probabilistic representation, we are able to construct a limit flow of maps on the probability measures. This result complements and improves the existing result coming from the classical sewing lemma. It is applied to the case of law-dependent jump SDEs for which we obtain weak existence result as well as the uniqueness of the marginal laws.