## Ricci curvature of Markov chains on metric spaces.(English)Zbl 1181.53015

Let $$x$$ and $$y$$ be two points in a Riemannian manifold $$M$$. Let $$w$$ be a tangent vector at $$x$$, let $$w'$$ be the tangent vector at $$y$$ obtained by parallel transport of $$w$$ from $$x$$ to $$y$$. If one follows the two geodesics issuing from $$x, w$$ and $$y,w'$$, in positive curvature the geodesics will get closer and will part away in negative curvature. Starting from this approach of Ricci curvature, the author notices that if one thinks of a direction $$w$$ at $$x$$ as a point on a small sphere $$S_x$$ centered at $$x$$, on average, the Ricci curvature controls wether the distance between a point of $$S_x$$ and the corresponding point of $$S_y$$, is smaller or larger than the distance from $$x$$ to $$y$$. Let $$(X,d)$$ be a polish metric space. In $$X$$ one can define a random walk as a family of probabilistic measures $$m_x(.)$$ on X for each $$x\in X$$satisfying some assumptions.
So now the idea is to replace the notion of “corresponding points” between two closed spheres by transportation distance between measures. The $$L^1$$ transportation distance between two probability measures $$\nu_1$$ and $$\nu_2$$ in $$X$$ will be given by $$W_1(\nu_1,\nu_2):=\inf \int_{(x,y)\in X\times X} d(x,y)d\xi (x,y)$$ and the infimum is taken over the set of measures in $$X\times X$$ projecting to $$\nu_1$$ and $$\nu_2$$ . The fact that the measures of the random walk are closer or further apart than the points $$x$$ and $$y$$ will give the sign of the Ricci curvature.
Coarse Ricci curvature $$\kappa (x,y)$$ is defined for a metric space $$(X,d)$$ with a random walk $$m$$. Let $$x$$ and $$y$$ be two points of $$X$$, then $$\kappa (x,y):=1-\frac{W_1(\nu_1,\nu_2)}{d(x,y)}$$.
According to the author “geometers will think of $$m_x$$ as a replacement for the notion of ball around $$x$$. Probabilists will rather think of this data as defining a Markov chain with a specific transition probability from $$x$$ to $$y$$”.
With this approach to Ricci curvature, the Ricci curvature is positive if and only if the random walk operator is contracting on the space of probability measure with the transportation metric. This generalization is consistent with the Bakry-Émery Ricci curvature for Brownian motion with a drift on a Riemannian manifold (see abstract).
The article includes an introduction that includes historical remarks and related work. It also includes one section with definitions and statements and one on elementary properties that will help the reader to get into this perspective since it includes lots of examples. The author proceeds with sections devoted to concentration results, local control and logarithmic Sobolev inequality, exponential concentration in non-negative curvature, $$L^2$$ Bonnet-Myers theorems, coarse Ricci curvature and Gromov-Hausdorff topology, and transportation distance in Riemannian manifolds.

### MSC:

 53B20 Local Riemannian geometry 54E35 Metric spaces, metrizability 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 60J99 Markov processes

Zbl 1132.53011
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### References:

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