  Multistability, i.e. the coexistence of several attractors for a given set of
system parameters is one of the most important phenomena occurring in dynamical
systems. We consider it in velocity dynamics of a Brownian particle driven by
thermal fluctuations and moving in a biased periodic potential. In doing so we
focus on the impact of ergodicity - a concept which lies at the core of
statistical mechanics. The latter implies that a single trajectory of the
system is representative for the whole ensemble and as a consequence the
initial conditions of the dynamics are fully forgotten. The ergodicity of the
deterministic counterpart is strongly broken and we discuss how the velocity
multistability depends on the starting position and velocity of the particle.
While for non-zero temperature the ergodicity is in principle restored, in the
low temperature regime the velocity dynamics is still affected by initial
conditions due to weak ergodicity breaking. For moderate and high temperature
the multistability is robust with respect to the choice of the starting
position and velocity of the particle.
