Asymptotic of the transport distance for Brownian occupation measures and random interlacements

In the last decades, some remarkable advances were achieved in the asymptotic of the so-called random matching problem. We focus on the case of continuous-time Markov processes, and discuss in particular the asymptotic of the transport distance between the occupation measure of a Brownian motion on a flat torus, and the Lebesgue measure. The result is established thanks to a comparison with an auxiliary random measure, called the random interlacement, for which the transport distance naturally features subadditivity. Possible generalizations and their connection with some conjectures that appeared in the physical literature will be further discussed, as time allows. Joint work with D.Trevisan.