Precompactness for abstract Bakry–Émery data

Establish a precompactness theorem for the space of abstract Bakry–Émery data, i.e., triples (A,H,L) equipped with a Markov semigroup and satisfying Bakry–Émery curvature-dimension inequalities, not necessarily arising from underlying metric-measure spaces.

Background

The Bakry–Émery framework provides an abstract calculus for Markov semigroups, capturing curvature-dimension conditions beyond smooth Riemannian settings.

While precompactness results are known for geometric classes (e.g., metric-measure spaces with lower Ricci bounds), a corresponding theorem for abstract data sets not tightly linked to a specific space is missing.