  We prove that random walks on Thompson's group $F$ driven by strictly
non-degenerate finitely supported probability measures $\mu$ have a non-trivial
Poisson boundary. The proof consists in an explicit construction of two
different non-trivial $\mu$-boundaries. Both of them are defined in terms of
the Schreier graph $\Gamma$ on the dyadic-rational orbit of the canonical
action of $F$ on the unit interval (actually, we consider a natural embedding
of $F$ into the group $PLF({\mathbb R})$ of piecewise linear homeomorphisms of
the real line, and realize $\Gamma$ on the dyadic-rational orbit in ${\mathbb
R}$). However, the behaviours at infinity described by these $\mu$-boundaries
are quite different (in perfect keeping with the ambivalence concerning
amenability of the group $F$). The first $\mu$-boundary is similar to the
boundaries of the lamplighter groups: it consists of ${\mathbb Z}$-valued
configurations on $\Gamma$ arising from the stabilization of the logarithmic
increments of slopes along the sample paths of the random walk. The second
$\mu$-boundary is more similar to the boundaries of groups with hyperbolic
properties as it consists of the sections of the end bundle of the graph
$\Gamma$: these are the collections of the limit ends of the induced random
walk on $\Gamma$ parameterized by all possible starting points.
