  Employing large deviation theory, we explore current fluctuations of
underdamped Brownian motion for the paradigmatic example of a single particle
in a one dimensional periodic potential. Two different approaches to the large
deviation function of the particle current are presented. First, we derive an
explicit expression for the large deviation functional of the empirical phase
space density, which replaces the level 2.5 functional used for overdamped
dynamics. Using this approach, we obtain several bounds on the large deviation
function of the particle current. We compare these to bounds for overdamped
dynamics that have recently been derived motivated by the thermodynamic
uncertainty relation. Second, we provide a method to calculate the large
deviation function via the cumulant generating function. We use this method to
assess the tightness of the bounds in a numerical case study for a cosine
potential.
