Two-sided Green function estimates for killed subordinate Brownian motions
A subordinate Brownian motion is a Lévy process that can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is \(-\phi (- \Delta )\), where \(\phi \) is the Laplace exponent of the subordinator.NEWLINENEWLINEIn this paper, the authors consider a large class of subordinate Brownian motions without diffusion component. The most important improvement with respect to their previous paper [Stochastic Processes Appl. 119, No. 5, 1601--1631 (2009; Zbl 1166.60046)] is that \(\phi \) is only supposed to be comparable (at infinity) to a regularly varying function. Thus, the estimates below hold for a lot more processes such as (sums of independent) symmetric stable processes, (sums of independent) relativistic stable processes. Also some redundant additional assumptions are eliminated.NEWLINENEWLINEThe main result gives the following sharp estimates for the Green function \(G_D \) of \(X\) in a bounded, \(\kappa \)-fat open set \(D \subset \mathbb R^d\): NEWLINE\[NEWLINE\begin{multlined} C_1 ^ {-1} \left (1 \wedge \frac{\phi (|x-y|^ {-2})}{\sqrt{\phi (\delta _D (x) ^ {-2} ) \phi (\delta _D (y) ^ {-2} ) }} \right ) \frac{1}{|x-y|^ {d} \phi (|x-y|^ {-2})}\\ \leq G_D (x, y) \leq C_1 \left (1 \wedge \frac{\phi (|x-y|^ {-2})}{\sqrt{\phi (\delta _D (x) ^ {-2} ) \phi (\delta _D (y) ^ {-2} ) }} \right ) \frac{1}{|x-y|^ {d} \phi (|x-y|^ {-2})}.\end{multlined}NEWLINE\]NEWLINE When \(D\) is moreover \(C^ {1, 1}\), the authors obtain an explicit form of the estimates in terms of the distance to the boundary. As a consequence, they obtain a boundary Harnack principle in \(C^ {1, 1}\) open sets with explicit rate of decay.
scientific article; zbMATH DE number 6042726