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\(\mathcal{L} = \tfrac{1}{2}\partial _{xx} - \alpha \partial _x\), starting from x>0 and T0 be the hitting time of 0. Consider the family of positive solutions of the equation\({\mathcal{L}}\psi = - \lambda \psi\) with λ∈(0, η), where\(\eta {\text{ = }} - \lim _{t \to \infty } (1/t){\text{ log }}\mathbb{P}_x (T_0 > t)\). We show that the distribution of the h-process induced by any such ψ is\(\lim _{M \to \infty } \mathbb{P}_x (X \in A|S_M < T_0 )\), for a suitable sequence of stopping times (SM : M≥0) related to ψ which converges to ∞ with M. We also give analytical conditions for \(\eta = \underline \lambda\), where \(\underline \lambda\) is the smallest point of increase of the spectral measure associated to \(\mathcal{L}^ *\).