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.