  We introduce a new condition on elliptic operators $L= {1/2}\triangle + b
\cdot \nabla $ which ensures the validity of the Liouville property for bounded
solutions to $Lu=0$ on $\R^d$. Such condition is sharp when $d=1$. We extend
our Liouville theorem to more general second order operators in non-divergence
form assuming a Cordes type condition.

Analysis of PDEs;Probability