Abstract
Let $$L=\Delta ^{\alpha /2}+ b\cdot \nabla $$ with $$\alpha \in (1,2)$$ . We prove the Martin representation and the Relative Fatou Theorem for non-negative singular L-harmonic functions on $$\mathcal{C }^{1,1}$$ bounded open sets.
Highlights
Martin representations and boundary properties of harmonic functions were widely studied in the case of diffusion operators
We prove the existence of the L-Martin kernel which is L-harmonic (Theorems 8 and 12)
We provide a proof of this theorem based on the perturbation formula for singular L-harmonic functions (29)
Summary
Analysis of harmonic functions related to fractional powers α/2 of the Laplace operator is an important topic, intensely developed in recent years, for. Where α ∈ (0, 2), constitute an important class of jump processes, intensely studied in recent years Their most celebrated case are the Ornstein–Uhlenbeck stable processes with b(x) = λx, λ ∈ R. The topics (i) and (ii) are addressed in this article and the subject (iii) in a forthcoming paper All these topics are fundamental for the knowledge of L-harmonic functions. Martin representations and boundary properties of harmonic functions were widely studied in the case of diffusion operators. The potential theory of stable stochastic processes with gradient perturbations was started in the Ornstein–Uhlenbeck case by Jakubowski [29,30]. Our work is a natural continuation of the research presented in [10]
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