Abstract

Let (Xt) be a one dimensional diffusion corresponding to the operator ? = 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 ?? = -?? with ??(0, ?), where ? = -limt??(1/t) log ?x(T0 > t). We show that the distribution of the h-process induced by any such ? is limM?? ?x(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 ?*.