Large deviations asymptotics for unbounded additive functionals of diffusion processes

Mihail Bazhba, Jose Blanchet, Roger J. A. Laeven, Bert Zwart

Published 2022-02-22Version 1

We study large deviations asymptotics for unbounded additive functionals of one-dimensional Langevin diffusions with sub-linear gradient drifts. Our results allow us to obtain 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 which dictates regions of sub-linear speed for our large deviations asymptotics. Our proof technique hinges on various constructions of independent interest, including a suitable decomposition of the diffusion process in terms of alternating renewal cycles and a decomposition of the paths on excursion. The key to the sub-linear behavior is a heavy-tailed large deviations phenomenon arising from the fact that at each regeneration cycle the accumulated area of the diffusion process displays a semi-exponential behavior.