Ricci curvature and gradient flows of the entropy for jump processes

In the first part of this thesis, we present a new notion of Ricci curvature that applies to finite Markov chains. This notion relies on geodesic convexity of the entropy and is analogous to the one introduced by Lott--Villani and Sturm for geodesic metric measure spaces. In order to apply to the discrete setting the role of the Wasserstein distance is taken over by a different metric W on the space of probability measures having the property that the continuous time Markov chain is the gradient flow of the entropy. Using this notion of Ricci curvature we prove discrete analogues of fundamental results by Bakry--Emery and Otto--Villani. In particular, we show that Ricci curvature bounds imply a number of functional inequalities for the invariant measure of the Markov chain. These include a modified logarithmic Sobolev inequality and a Talagrand-type transport inequality involving the distance W. Moreover, we prove that Ricci curvature bounds are stable under tensorisation. As a special case we obtain the sharp Ricci curvature bound for the simple random walk on the discrete hypercube. In the second part, we take a similar approach towards jump processes on R^d. We introduce a new transport distance W between probability measures on R^d that is built from a jump kernel of Lévy measures. It is defined via a non-local variant of the dynamical characterisation of the Wasserstein distance. We study geometric and topological properties of the distance W. In particular, we prove that every pair of probability measures at finite distance can be connected by a geodesic. We put particular focus on translation invariant jump kernels and consider the associated non local operator which is the generator of a pure jump Lévy process. We prove that the semigroup generated by this non-local operator is the gradient flow of the relative entropy with respect to the distance W. This is reminiscent of the Jordan--Kinderlehrer--Otto interpretation of the heat equation as the gradient flow of the entropy w.r.t. the Wasserstein distance. Moreover, we show that the entropy is convex along W-geodesics. As a special case, we obtain a gradient flow characterisation of the semigroup generated by the fractional Laplacian