What do you mean by the Liouville theorem? If the absence of bounded or positive harmonic functions, then the answer is "no" due to the presence of a vector field $b$. The corresponding counterexample can be constructed already for $n=1$. Take the diffusion coefficient $a=a_{11}$ to be equal identically 1, and let $b=b_1$ be odd and such that it converges to $+1$ at $+\infty$ (and therefore to $-1$ at $-\infty$). Then the space of bounded harmonic functions is 2-dimensional. This can be seen either directly, or from probabilistic considerations: the corresponding diffusion process on $\mathbb R$ converges to one of the two ends of the real line, so that it has a non-trivial behaviour at infinity.