  Let $G$ be a locally compact group and $E$ be a $G$-space. An irreducible
probability measure $\mu$ on $G$ is said to have Liouville property on $E$ if
$G$-invariant functions on $E$ are the only continuous bounded functions on $E$
that satisfy the mean value property with respect to $\mu$. We first prove that
the random walk induced by $\mu$ on $E$ is transient outside a closed set and
on the closed set $\mu$ has Liouville. We mainly consider actions on vector
spaces and projective spaces. We show that measures on $GL(V)$ that are
supported inside a ball of radius less than $a<1$ have Liouville property on
$V$. We also prove that measures on $GL(\R ^2)$ have Liouville property on the
projective line. We next exhibit subgroups of $GL(V)$ so that irreducible
measures on such subgroups have Liouville on the projective space $\mP (V)$ of
$V$. We also prove irreducible measures on $SL(V)$ have Liouville property on
$\mP (\cSL (V))$ where $\cSL (V)$ is the Lie algebra of $SL(V)$
