Rates of Decay and h-Processes for One Dimensional Diffusions Conditioned on Non-Absorption

Abstract

Let (Xt ) be a one dimensional diffusion corresponding to the operator L= 1 2 xx&: x , starting from x>0 and T0 be the hitting time of 0. Consider the family of positive solutions of the equation L =&* with * # (0, '), where '=&limt (1 t) log Px(T0>t). We show that the distribution of the h-process induced by any such is limM Px(X # A | SM<T0), for a suitable sequence of stopping times (SM :M 0) related to which converges to with M. We also give analytical conditions for '=*

, where*

is the smallest point of increase of the spectral measure associated to L*.