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Theses

Courbure de Ricci grossière de processus markoviens

Abstract : The coarse Ricci curvature of a Markov process on a Polish space is defined as a local contraction rate of the W1 Wasserstein distance between the laws of the process starting at two different points. The first part of this thesis deals with results holding in the case of general Polish spaces. The simplest of them is that the infimum of the coarse Ricci curvature is a global contraction rate of the semigroup of the process for the W1 distance between probability measures. Though intuitive, this result is diffucult to prove in continuous time. The proof of this result, and the following consequences for the spectral gap of the generator are the subject of Chapter 1. Another interesting result, using the values of the coarse Ricci curvature at different points, and not only its infimum, is a concentration result for the equilibrium measures, only holding in a discrete time framework. That will be the topic of Chapter 2. The second part of this thesis deals with the particular case of diffusions on Riemannian manifolds. A formula is given, allowing to get the coarse Ricci curvature from the generator of the diffusion. In the case when the metric is adapted to the diffusion, we show the existence of a coupling between the paths starting at two different points, such that the coarse Ricci curvature is exactly the decreasing rate of the distance between these paths. We can then show that the spectral gap of the generator is at least the harmonic mean of the Ricci curvature. This result can be generalized when the metric is not the one induced by the generator, but it needs a very restricting hypothesis, and the curvature we have to choose is smaller.
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Laurent Veysseire. Courbure de Ricci grossière de processus markoviens. Mathématiques générales [math.GM]. Ecole normale supérieure de lyon - ENS LYON, 2012. Français. ⟨NNT : 2012ENSL0732⟩. ⟨tel-00737099⟩

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