Scaling Limits for Conditional Diffusion Exit Problems and Asymptotics for Nonlinear Elliptic Equations

The goal of this paper is to supplement the large deviation principle of the Freidlin–Wentzell theory on exit problems for diffusion processes with results of classical central limit theorem kind. Namely, we describe a class of situations where conditioning on exit through unlikely locations leads to a Gaussian scaling limit for the exit distribution. Our results are based on Doob’s h-transform and new asymptotic convergence gradient estimates for elliptic nonlinear equations that allow one to reduce the problem to the Levinson case. We devote an appendix to a rigorous and general discussion of h-transform.