Recurrent Perturbations of Certain Transient Radially Symmetric Diffusions

Abstract

If $L$ generates a transient diffusion, then the corresponding exterior Dirichlet problem (EP) has in general many bounded solutions. We consider perturbations of $L$ by a first-order term and assume that EP can be solved uniquely for each perturbed operator. Then as the perturbation tends to 0, the sequence of perturbed solutions may converge to a solution of the original EP. Using a skew-product representation of diffusions, we give an integral criterion for the uniqueness of this limit and show that it takes place iff the Kuramochi boundary of $L$ at $\infty$ is a singleton. In the case when uniqueness fails, we provide a description of a subclass of limiting solutions in terms of boundary conditions for the original process in the natural scale.