**4. Discussion**

In conclusion, we thoroughly investigated the influence of initial conditions distribution on the multistability of velocity dynamics for the Brownian particle in a tilted periodic potential. The ergodicity of the deterministic system is strongly broken, and therefore the initial conditions are never forgotten and crucially impact the obtained results. The only way to correctly sample the whole state space of the system is to average over them in such a way that no single one is preferred. The latter condition translates to a uniform distribution in the initial phase space of the system. We demonstrated that, while for non-zero temperatures the ergodicity is in principle restored, in a low temperature regime the results are still significantly affected by the initial conditions due to the weak ergodicity breaking. This means that the time needed for the ergodicity reinstatement tends to infinity when temperature goes down to zero. For moderate and high temperature regimes, the detected multistability is robust with respect to the choice of initial conditions. It is valid even when the whole ensemble starts from a given point in the phase space of the system. A remaining question is how the time needed for the ergodicity restoration depends on temperature. This constitutes a challengeable objective which we hope to address in the future.

**Author Contributions:** Conceptualization, J.S.; Investigation, J.S.; Software, J.S.; Visualization, J.S.; Writing—original draft, J.S.; Writing—review & editing, J.S., P.H. and J.Ł. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has been supported by the Grant NCN No. 2017/26/D/ST2/ 00543 (J.S.).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data that support the findings of this study are available from the corresponding author upon reasonable request.

**Acknowledgments:** P.H. and J.S. wish to acknowledge the financial support obtained via the University of Augsburg Guestprogramme funded by Bavarian Ministery for Science and Art, Germany.

**Conflicts of Interest:** The authors declare no conflict of interest.
