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.
Citation
D. Ioffe. "Recurrent Perturbations of Certain Transient Radially Symmetric Diffusions." Ann. Probab. 21 (2) 1124 - 1150, April, 1993. https://doi.org/10.1214/aop/1176989284
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