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The goal of this paper is to supplement the large deviation principle of theFreidlin--Wentzell theory on exit problems for diffusion processes with resultsof classical central limit theorem kind. We describe a class of situationswhere conditioning on exit through unlikely locations leads to a Gaussianscaling limit for the exit distribution. Our results are based on Doob'sh-transform and new asymptotic convergence gradient estimates for ellipticnonlinear equations that allow to reduce the problem to the Levinson case. Wedevote a separate section to a rigorous and general discussion of h-transform.

Scaling limits for conditional diffusion exit problems, Doob's h-transform, and asymptotics for nonlinear elliptic equations