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We give sufficient conditions for the non-triviality of the Poisson boundaryof random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$is the group of piecewise projective homeomorphisms over the integers definedby Monod. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$, we provethat either $H$ is solvable, or every measure on $H$ with finite first momentthat generates it as a semigroup has non-trivial Poisson boundary. Inparticular, we prove the non-triviality of the Poisson boundary of measures onThompson's group $F$ that generate it as a semigroup and have finite firstmoment, which answers a question by Kaimanovich.

Non-triviality of the Poisson boundary of random walks on the group $H(\mathbb{Z})$ of Monod