Let us consider an infinite continuous or discrete metric space $(S,d)$ equipped with a probability measure $\mu$.  For instance a Riemannian manifold with a potential,  such as the sphere equipped with its uniform measure, or $\mathbb{R}^n$ equipped with the Gaussian measure. One of the most studied problems are the concentration of measure phenomenon for $\mu$  and the numerical simulation of $\mu$. To address these problems, one may introduce an auxiliary Markov process  $(X_t)_{t\geq0}$  on $S$ which admits $\mu$ as an invariant distribution (e.g. the Brownian motion on the sphere and the Ornstein-Uhlenbeck process on $\mathbb{R}^n$). Perhaps the most famous instance of the approach is the Metropolis-Hastings simulation algorithm (Monte Carlo Markov Chain).
The speed of convergence of  $(X_t)_{t\geq0}$ to $\mu$ in entropy can be related to the concentration of measure of $\mu$. In the case of a Riemannian manifold, this can be related to a notion of curvature taking into account both the geometry (Ricci) of the manifold and the Markov dynamics. It is known as the Bakry-Émery curvature. By using Wasserstein coupling instead of entropy, Ollivier extended this notion of curvature to discrete spaces (there are some connections with ideas of M.-F. Chen, Joulin, Villani, Lott, and Sturm). This allows primarily to control the concentration of measure of $\mu$. From this point of view the Markov process is instrumental.