We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on
$H(\mathbb{Z})$
and its subgroups. The group
$H(\mathbb{Z})$
is the group of piecewise projective homeomorphisms over the integers defined by Monod [Groups of piecewise projective homeomorphisms. Proc. Natl Acad. Sci. USA110(12) (2013), 4524–4527]. For a finitely generated subgroup
$H$
of
$H(\mathbb{Z})$
, we prove that either
$H$
is solvable or every measure on
$H$
with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson’s group
$F$
that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich [Thompson’s group
$F$
is not Liouville. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series). Eds. T. Ceccherini-Silberstein, M. Salvatori and E. Sava-Huss. Cambridge University Press, Cambridge, 2017, pp. 300–342, 7.A].