  Given a finitely generated group, the well-known Stability Problem asks
whether the non-triviality of the Poisson-Furstenberg boundary (which is
equivalent to the existence of non-constant bounded harmonic functions) depends
on the choice of simple random walk on the group. This question was far from
being understood even in the class of linear groups. Given an amenable group,
e.g. a solvable group, there is no known characterisation, even a conjectural
one, of when it admits a simple random walk with non-trivial boundary. We
provide a characterisation of groups with non-trivial boundary for finitely
generated linear groups of characteristic $p$. We prove in particular that the
Stability Problem has a positive answer in this class of groups. For linear
groups of characteristic $0$, we prove a sufficient condition for the
triviality of the boundary which does not depend on the choice of a simple
random walk. We conjecture that our sufficient condition is also necessary. Our
arguments are based on a new comparison criterion for group extensions, on new
$\Delta$-restriction entropy estimates and a criterion for boundary
non-triviality, and on a new "cautiousness" criterion for triviality of the
boundary.
