  It is well-known that Brownian ratchets can exhibit current reversals,
wherein the sign of the current switches as a function of the driving
frequency. We introduce a spatial discretization of such a two-dimensional
Brownian ratchet to enable spectral methods that efficiently compute those
currents. These discrete-space models provide a convenient way to study the
Markovian dynamics conditioned upon generating particular values of the
currents. By studying such conditioned processes, we demonstrate that
low-frequency negative values of current arise from typical events and
high-frequency positive values of current arises from rare events. We
demonstrate how these observations can inform the sculpting of time-dependent
potential landscapes with a specific frequency response.
