  We show under weak hypotheses that $\partial X$, the Roller boundary of a
finite dimensional CAT(0) cube complex $X$ is the Furstenberg-Poisson boundary
of a sufficiently nice random walk on an acting group $\Gamma$. In particular,
we show that if $\Gamma$ admits a nonelementary proper action on $X$, and $\mu$
is a generating probability measure of finite entropy and finite first
logarithmic moment, then there is a $\mu$-stationary measure on $\partial X$
making it the Furstenberg-Poisson boundary for the $\mu$-random walk on
$\Gamma$. We also show that the support is contained in the closure of the
regular points. Regular points exhibit strong contracting properties.
