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Liouville property on $G$-spaces

Let $G$ be a locally compact group and $E$ be a $G$-space. An irreducible probability measure $μ$ 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 $μ$. We first prove that the random walk induced by $μ$ on $E$ is transient outside a closed set and on the closed set $μ$ 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 $SL(V)$