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 deffererential)xx- (alpha)(partial defferential)xx, starting from x>0 and T 0 be the hitting time of 0. Consider the family of positive solutions of the equation l(psi)=-(lambda)(psi)with (lambda)(element of)(0, (eta)), where (eta)=-lim t-(infinity)(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-(infinity) Px(X(element of)A|S m=0) related to (psi)which converges to (infinity)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*.