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We study large deviations asymptotics for a class of unbounded additivefunctionals, interpreted as normalized accumulated areas, of one-dimensionalLangevin diffusions with sub-linear gradient drifts. Our results provideparametric insights on the speed and the rate functions in terms of the growthrate of the drift and the growth rate of the additive functional. We find acritical value in terms of these growth parameters that dictates regions ofsub-linear speed for our large deviations asymptotics. Our approach is basedupon various constructions of independent interest, including a decompositionof the diffusion process in terms of alternating renewal cycles and a detailedanalysis of the paths during a cycle using suitable time and spatial scales.The key to the sub-linear behavior is a heavy-tailed large deviationsphenomenon arising from the principle of a single big jump coupled with theresult that at each regeneration cycle the upper-tail asymptotic behavior ofthe accumulated area of the diffusion process is proven to be semi-exponential(i.e., of heavy-tailed Weibull type).

Large deviations asymptotics for unbounded additive functionals of diffusion processes