Transport-Entropy Inequalities and Curvature in Discrete-Space Markov Chains

Abstract

Let $G = (\Omega,E)$ be a graph and let d be the graph distance. Consider a discrete-time Markov chain {Z _t } on $\Omega$ whose kernel p satisfies p(x, y) > 0 ⇒ {x, y} ∈ E for every $x,y \in \Omega$. In words, transitions only occur between neighboring points of the graph. Suppose further that $(\Omega,p,d)$ has coarse Ricci curvature at least 1∕α in the sense of Ollivier: For all $x,y \in \Omega$, it holds that $W_{1}(Z_{1}\mid \{Z_{0} = x\},Z_{1}\mid \{Z_{0} = y\}) \leq \left (1 -\frac{1} {\alpha } \right )d(x,y),$ where W ₁ denotes the Wasserstein 1-distance.

In this note, we derive a transport-entropy inequality: For any measure μ on $\Omega$, it holds that

$$\displaystyle{W_{1}(\mu,\pi ) \leq \sqrt{ \frac{2\alpha } {2 - 1/\alpha }D\!\left (\mu \,\|\,\pi \right )}\,,}$$

where π denotes the stationary measure of {Z _t } and $D\!\left (\cdot \,\|\,\cdot \right )$ is the relative entropy.

Peres and Tetali have conjectured a stronger consequence of coarse Ricci curvature, that a modified log-Sobolev inequality (MLSI) should hold, in analogy with the setting of Markov diffusions. We discuss how our approach suggests a natural attack on the MLSI conjecture.