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Rates of Decay and h-Processes for One Dimensional Diffusions Conditioned on Non-Absorption

Authors:	Servet Martínez Jaime San Martín

Institution:	(1) Facultad de Ciencias Físicas y Matemáticas, Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170-3, Correo 3, Santiago, Chile

Abstract:	Let (X _t) be a one dimensional diffusion corresponding to the operator $\mathcal{L} = \tfrac{1}{2}\partial _{xx} - \alpha \partial _x$ , starting from x>0 and T ₀ be the hitting time of 0. Consider the family of positive solutions of the equation ${\mathcal{L}}\psi = - \lambda \psi$ with (0, ), 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 is $\lim _{M \to \infty } \mathbb{P}_x (X \in A\|S_M < T_0 )$ , for a suitable sequence of stopping times (S _M : M0) related to 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 $\mathcal{L}^ *$ .

Keywords:	one-dimensional diffusions h-processes absorption
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