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Journal Article

# Asymptotic Laws for One-Dimensional Diffusions Conditioned to Nonabsorption

Pierre Collet, Servet Martinez and Jaime San Martin
The Annals of Probability
Vol. 23, No. 3 (Jul., 1995), pp. 1300-1314
Stable URL: http://www.jstor.org/stable/2244874
Page Count: 15

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## Abstract

If (Xt) is a one-dimensional diffusion corresponding to the operator L = 1/2∂xx - α∂x starting from $x > 0$ and Ta 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 Lh = - η 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 η can only take two values: η = 0 or $\eta = \underline{\lambda}$, where $\underline{\lambda}$ is the smallest point of increase of the spectral distribution of the operator L* = 1/2∂xx + ∂x(α·).

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