The classical Liouville property asserts that bounded harmonic functions on Euclidean space are necessarily constant. This property has been extended to $\mu$-harmonic functions related to a random walk $S$ in a locally compact group $G$ with defining measure $\mu$. In the present talk the dependence on $G$ and $\mu$, of the asymptotic entropy $h(G,\mu)$ of $S$, will be studied. The case $h(G,\mu)=0$ characterizes the Liouville property, and $h(G,\mu)>0$ leads to the well-known boundary theory of H. Furstenberg.