The small noise exit problem of Wentzell and Freidlin is of particular interest for regions whose boundary consists of trajectories of the underlying (unperturbed) dynamical system. This is called the case of characteristic boundary. One fruitful approach to attacking this problem involves conditioning the probability measure so that exit to the boundary occurs more quickly. In a previous paper, this approach was applied to some simplified examples, revealing some previously unanticipated phenomena for the characteristic boundary exit problem. In this paper we develop certain aspects of this approach more generally. In particular we present stochastic differential equations which give an asymptotically correct description of this conditioned process by using a carefully chosen system of coordinates near the boundary.
"Conditional Exits for Small Noise Diffusions with Characteristic Boundary." Ann. Probab. 20 (3) 1385 - 1419, July, 1992. https://doi.org/10.1214/aop/1176989696