  We reinvestigate a paradigmatic model of nonequilibrium statistical physics
consisting of an inertial Brownian particle in a symmetric periodic potential
subjected to both a time--periodic force and a static bias. In doing so we
focus on the negative mobility phenomenon in which the average velocity of the
particle is opposite to the constant force acting on it. Surprisingly, we find
that in the weak dissipation regime thermal fluctuations induce negative
mobility much more frequently than it happens if the dissipation is stronger.
In particular, for the very first time we report a parameter set in which
thermal noise causes this effect in the nonlinear response regime. Moreover, we
show that the coexistence of deterministic negative mobility and chaos is
routinely encountered when approaching the overdamped limit in which chaos does
not emerge rather than near the Hamiltonian regime of which chaos is one of the
hallmarks. On the other hand, at non-zero temperature the negative mobility in
the weak dissipation regime is typically affected by the weak ergodicity
breaking. Our findings can be corroborated experimentally in a multitude of
physical realizations including e.g.~Josephson junctions and cold atoms
dwelling in optical lattices.
