Liouville theorem for bounded harmonic functions on manifolds and graphs satisfying non-negative curvature dimension condition

Brighton [Bri13] proved the Liouville theorem for bounded harmonic functions on weighted manifolds satisfying non-negative curvature dimension condition, i.e. $CD(0,\mathrm{\infty })\mathrm{.}$ In this paper, we provide a new proof of this result by using the reverse Poincar'e inequality. Moreover, we adopt this approach to prove the Liouville theorem for bounded harmonic functions on graphs satisfying the $CD(0,\mathrm{\infty })$ condition.