  We consider a particle with a Langevin dynamics driven by a uniform
non-conservative force, in a one-dimensional potential with periodic boundary
conditions. We are interested in the properties of the system for atypical
values of the time-integral of a generalized particle current. To study these,
we bias the dynamics, at trajectory level, by a parameter conjugated to the
current, within the large-deviation formalism. We investigate, in the
weak-noise limit, the phase diagram spanned by the physical driving force and
the parameter defining the biased process. We focus in particular on the
depinning transition in this two-dimensional phase diagram. In the absence of
trajectory bias, the depinning transition as a function of the force is
characterized by the standard exponent $\frac{1}{2}$. We show that for any
non-zero bias, the depinning transition is characterized by an inverse
logarithmic behavior as a function of either the bias or the force, close to
the critical lines. We also report a scaling exponent $\frac{1}{3}$ for the
current when considering the depinning transition in terms of the bias, fixing
the non-conservative force to its critical value in the absence of bias. Then,
focusing on the time-integrated particle current, we study the thermal rounding
effects in the zero-current phase when the tilted potential exhibits a local
minimum. We derive in this case the Arrhenius scaling, in the small noise
limit, of both the particle current and the scaled cumulant generating
function. This derivation of the Arrhenius scaling relies on the determination
of the left eigenvector of the biased Fokker-Planck operator, to exponential
order in the low-noise limit. An effective Poissonian statistics of the
integrated current emerges in this limit.
