Rates of decay and h-processes for one dimensional diffusions conditioned on non-absorption

Let (X-t) be a one dimensional diffusion corresponding to the operator L = (1)/(2)partial derivative (xx) - alpha partial derivative (x), starting from x > 0 and T-0 be the hitting time of 0. Consider family of positive solutions of the equation L-psi = -lambda psi with lambda epsilon (0, eta), where eta = -lim(t --> proportional to)(1 . t) log P-x(T-0 > t). We show that the distribution of the h-process induced by any such psi is lim(M --> proportional to) P-x(X epsilon A \ S-M < T-0), for a suitable sequence of stopping times (S-M : M <greater than or equal to> 0) related to psi which coverages to proportional to with M. We also give analytical conditions for eta = lambda, where lambda is the smallest point of increase of the spectral measure associated to L*.