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Renewal Theorems for Singular Differential Operators

Authors:	Léonard Gallardo Khalifa Trimèche

Institution:	(1) Faculté des Sciences et Techniques Département de Mathématiques, Université de Tours, Parc de Grandmont, 37200 Tours, France;(2) Département de Mathématiques, Campus Universitaire, Faculté des Sciences de Tunis, 1060 c[Tunis, Tunisie

Abstract:	Let * be the convolution on M( ${\mathbb{R}}$ ₊) associated with a second order singular differential operator L on ]0, +. If is a probability measure on ${\mathbb{R}}$ ₊ with suitable moment conditions, we study how to normalize the measures ⁿ; n ${\mathbb{N}}$ } (resp. $\left\{ {\varepsilon _x \sum _{n = 0}^\infty \mu ^{ * n} } \right\}$ ) in order to get vague convergence if n+ (resp. x+). The results depend on the asymptotic drift of the operator L and on a precise study of the asymptotic behaviour of its eigenfunctions.

Keywords:	renewal theorems Laplace operator potential measure eigenfunctions vague convergence
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