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 ${\mathrm{\Delta }}_{X}u=0$, where $X$ is a vector field on a complete, non-compact Riemannian manifold and ${\mathrm{\Delta }}_X$ is the drifted Laplacian. In particular, we show that if the $X$-Bakry-Émery-Ricci curvature ${\mathrm{R}\mathrm{i}\mathrm{c}}_X$ 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 ${\mathrm{\Delta }}_{X}u+F(u)=0$, which holds when $F$ satisfies the structural conditions $t{F}^{\mathrm{\prime }}(t)-F(t)\le \alpha$ and $\mathrm{\mid }F(t)\mathrm{\mid }\le \beta t$, and the manifold has ${\mathrm{R}\mathrm{i}\mathrm{c}}_X\ge -(n-1)K$.