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July, 1995 Asymptotic Laws for One-Dimensional Diffusions Conditioned to Nonabsorption
Pierre Collet, Servet Martinez, Jaime San Martin
Ann. Probab. 23(3): 1300-1314 (July, 1995). DOI: 10.1214/aop/1176988185


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 $\mathscr{L}^\ast = \frac{1}{2}\partial_{xx} + \partial_x(\alpha\cdot)$.


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Pierre Collet. Servet Martinez. Jaime San Martin. "Asymptotic Laws for One-Dimensional Diffusions Conditioned to Nonabsorption." Ann. Probab. 23 (3) 1300 - 1314, July, 1995.


Published: July, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0867.60046
MathSciNet: MR1349173
Digital Object Identifier: 10.1214/aop/1176988185

Primary: 60J60
Secondary: 60F99

Keywords: $h$-processes , absorption , one-dimensional diffusions

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 3 • July, 1995
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