Bôcher type theorem for elliptic equations with drift perturbed Lévy operator
Tomasz Klimsiak
TL;DR
This work extends the classical Bôcher theorem to positive solutions of drift-perturbed Lévy equations by developing a probabilistic potential theory framework. The authors introduce a general setting where the Lévy operator $A$ with drift $b$ is augmented by a Radon measure $\lambda$, and they require the resolvent to be strongly Feller, enabling a robust decomposition of positive solutions through Radon measures and polar sets. The main results establish (i) a distributional representation $-Au+b\cdot\nabla u+\lambda=\mu_0+\sigma_K$, (ii) a finely continuous version with an explicit local Green-function expansion, and (iii) a maximum-principle-type inequality controlling the solution via boundary data. The approach covers subcritical and supercritical regimes for fractional and more general Lévy operators, including mixed and degenerate-diffusion cases, and it yields a versatile toolkit for analyzing isolated singularities in nonlocal and local settings with potential applications to fractional PDEs and stochastic processes.
Abstract
A classical Bôcher's theorem asserts that any positive harmonic function (with respect to the Laplacian) in the punctured unit ball can be expressed, up to the multiplication constant, as the sum of the Newtonian kernel and a positive function that is harmonic in the whole unit ball. This theorem expresses one of the fundamental results in the theory of isolated singularities and it can be viewed as a statement on the asymptotic behavior of positive harmonic functions near their isolated singularities. In the paper we generalize this results to drift perturbed Lévy operators. We propose a new approach based on the probabilistic potential theory. It applies to Lévy operators for which the resolvent of its perturbation is strongly Feller. In particular our result encompasses drift perturbed fractional Laplacians with any stability index bounded between zero and two - the method therefore applies to subcritical and supercritical cases.