Renewal Theorems for Singular Differential Operators

Let * be the convolution on M(R+) associated with a second order singular differential operator L on ]0, +(infinity)[. If (mu)is a probability measure on R+ with suitable moment conditions, we study how to normalize the measures (mu)* n ; n(element of) N} (resp.{(epsilon) x * (sigma)=0(mu)^*n}) in order to get vague convergence if n-+(infinity) (resp. x-+(infinty)). The results depend on the asymptotic drift of the operator L and on a precise study of the asymptotic behaviour of its eigenfunctions.