Green function estimates for relativistic stable processes in half-space-like open sets
Let \(X_t^m\) be the relativistic \(\alpha\)-stable process in \(\mathbb R^n\), i.e., the process with characteristic exponent \((|\xi|^2+m^2)^{\frac{\alpha}{2}}-m\), \(m>0\), \(X_t\equiv X_t^0\). Denote by \(X_t^{m,D}\) (resp., \(X_t^D\)) the process killed upon exiting the domain \(\mathcal{D}\). The paper is devoted to sharp two-sided estimates for the Green function related to \(X_t^D\) when \(\mathcal{D}\) is a half-space-like \(C^{1,1}\) open set. For the case when \(\mathcal{D}\) is a \(C^{1,1}\)-open set in \(\mathbb R^n\), sharp estimates for the transition probability \(p_t^{m,D}(t,x,y)\) and the Green function \(G_D^m(x,y)\) were given in [the authors, Ann. Probab. 40, No.~1, 213--244 (2012; Zbl 1235.60101)]. For bounded \(\mathcal{D}\), the estimates for the Green function show that it is comparable to the Green function of the killed stable process in \(\mathcal{D}\). In this paper, the estimates for the Green function are obtained under the assumption that \(\mathcal{D}\) is an unbounded \(C^{1,1}\) domain.
scientific article; zbMATH DE number 5908028
scientific article