Gradient estimates and Liouville properties for the drifted Laplacian

In this paper, we discuss the validity of the Liouville property for $X$-harmonic functions, i.e. positive solution to $\Delta_{X}u=0$, where $X$ is a vector field on a complete, non-compact Riemannian manifold and $\Delta_{X}$ is the drifted Laplacian. In particular, we show that if the $X$-Bakry-\'Emery-Ricci curvature $\mathrm{Ric}_{X}$ is non-negative and the norm of $X$ decays to zero at infinity, then the manifold has the Liouville property for the $X$-Laplacian. The proof exploits a local gradient estimate for positive solutions to the semilinear equation $\Delta_{X}u+F(u)=0$, which holds when $F$ satisfies the structural conditions $tF'(t)-F(t)\le\alpha$ and $\vert F(t)\vert\le\beta t$, and the manifold has $\mathrm{Ric}_{X}\ge-(n-1)K$.