Entropy-Transport distances between unbalanced metric measure spaces

Inspired by the recent theory of Entropy-Transport problems and $\mathbf{D}$-distance Sturm on normalised metric measure spaces, we define a new class complete separable distances between spaces possibly different total mass. We provide several explicit examples such distances, where prominent role is played geodesic based Hellinger-Kantorovich distance. Moreover, discuss some limiting cases theory, recovering "pure transport" introducing entropic" distances. also study in detail topology induced metrics, showing compactness stability results for satisfying Ricci curvature lower bounds synthetic sense.