  We study large deviations asymptotics for a class of unbounded additive
functionals, interpreted as normalized accumulated areas, of one-dimensional
Langevin diffusions with sub-linear gradient drifts. Our results provide
parametric insights on the speed and the rate functions in terms of the growth
rate of the drift and the growth rate of the additive functional. We find a
critical value in terms of these growth parameters that dictates regions of
sub-linear speed for our large deviations asymptotics. Our approach is based
upon various constructions of independent interest, including a decomposition
of the diffusion process in terms of alternating renewal cycles and a detailed
analysis of the paths during a cycle using suitable time and spatial scales.
The key to the sub-linear behavior is a heavy-tailed large deviations
phenomenon arising from the principle of a single big jump coupled with the
result that at each regeneration cycle the upper-tail asymptotic behavior of
the accumulated area of the diffusion process is proven to be semi-exponential
(i.e., of heavy-tailed Weibull type).
