Least gradient functions in metric random walk spaces

In this paper we study least gradient functions in metric random walk spaces, which include as particular cases the least gradient functions on locally finite weighted connected graphs and nonlocal least gradient functions on $\mathbb{R}^N$. Assuming that a Poincar\'e inequality is satisfied, we study the Euler-Lagrange equation associated with the least gradient problem. We also prove the Poincar\'e inequality in a few settings.