At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and
John Mather launched a revolution in the venerable field of optimal transport founded by G …

In this paper, we introduce a synthetic notion of Riemannian Ricci bounds from below for
metric measure spaces (X, d, m) which is stable under measured Gromov–Hausdorff …

We prove the equivalence of the curvature-dimension bounds of Lott–Sturm–Villani (via
entropy and optimal transport) and of Bakry–Émery (via energy and Γ _2 Γ 2-calculus) in …

The theory of curvature-dimension bounds for nonsmooth spaces has several motivations:
the study of functional and geometric inequalities in structures which arc very far from being …

The aim of the present paper is to bridge the gap between the Bakry–Émery and the Lott–
Sturm–Villani approaches to provide synthetic and abstract notions of lower Ricci curvature …

The aim of this paper is to provide new characterizations of the curvature dimension
condition in the context of metric measure spaces $(X,\mathsf {d},\mathfrak {m}) $. On the …

We prove that a metric measure space (X, d, m) satisfying finite-dimensional lower Ricci
curvature bounds and whose Sobolev space W1, 2 is Hilbert is rectifiable. That is, an …

This survey gives an account of the state of the art of the theory of Ornstein–Uhlenbeck
operators and semigroups. The domains of definition and the spectra of such operators are …

We prove that if (X, d, m)(X, d, m) is a metric measure space with m (X)= 1 m (X)= 1 having
(in a synthetic sense) Ricci curvature bounded from below by K> 0 K> 0 and dimension …

Let L= Δ−∇ ϕ⋅∇ be a symmetric diffusion operator with an invariant measure μ (dx)= e− ϕ
(x) dx on a complete non-compact Riemannian manifold M. We give the optimal conditions …