Dimensional contraction via Markov transportation distance

François Bolley (1) , Ivan Gentil (2) , Arnaud Guillin (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 EDPA - Équations aux dérivées partielles, analyse
3 LMBP - Laboratoire de Mathématiques Blaise Pascal

It is now well known that curvature conditions à la Bakry-Emery are equivalent to contraction properties of the heat semigroup with respect to the classical quadratic Wasserstein distance. However, this curvature condition may include a dimensional correction which up to now had not induced any strenghtening of this contraction. We first consider the simplest example of the Euclidean heat semigroup, and prove that indeed it is so. To consider the case of a general Markov semigroup, we introduce a new distance between probability measures, based on the semigroup, and adapted to it. We prove that this Markov transportation distance satisfies the same properties for a general Markov semigroup as the Wasserstein distance does in the specific case of the Euclidean heat semigroup, namely dimensional contraction properties and Evolutional variational inequalities.

Mots clés: Wasserstein distance; Diffusion equations; Markov semigroups; Curvature-dimension bounds

Domaines: Probabilités [math.PR]; Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

François Bolley, Ivan Gentil, Arnaud Guillin. Dimensional contraction via Markov transportation distance. Journal of the London Mathematical Society, 2014, 90 (1), pp.309-332. ⟨hal-00808717v3⟩