Asymptotic Laws for One-Dimensional Diffusions Conditioned to Nonabsorption

Abstract

If $(X_t)$ is a one-dimensional diffusion corresponding to the operator $\mathscr{L} = \frac{1}{2}\partial_{xx} - \alpha\partial_x$ starting from $x > 0$ and $T_a$ is the hitting time of $a$, we prove that under suitable conditions on the drift coefficient the following limit exists: $\forall s > 0, \forall A \in \mathscr{F}_s, \lim_{t\rightarrow\infty} \mathbb{P}_x(X \in A\mid T_0 > t)$. We characterize this limit as the distribution of an $h$-like process, $h$ satisfying $\mathscr{L}h = - \eta h, h(0) = 0, h'(0) = 1$, where $\eta = -\lim_{t\rightarrow\infty}(1/t)\log\mathbb{P}_x(T_0 > t)$. Moreover, we show that this parameter $\eta$ can only take two values: $\eta = 0$ or $\eta = \underline{\lambda}$, where $\underline{\lambda}$ is the smallest point of increase of the spectral distribution of the operator $\mathscr{L}^\ast = \frac{1}{2}\partial_{xx} + \partial_x(\alpha\cdot)$.