On some path properties of symmetric stable-like processes for one dimension
The author uses Dirichlet form techniques in order to study stable-like processes on \({\mathbb R}^d\). More precisely, for a measurable function \(\alpha\) on \({\mathbb R}^d\) let \[ {\mathcal E}^{\alpha}(u,v)=\iint_{{\mathbb R}^d\times {\mathbb R}^d} {(u(x)-(u(y))^2\over |x-y|^{d+\alpha(x)}} dx dy \] be a symmetric quadratic form on \(L^2({\mathbb R}^d; dx)\) with the domain \({\mathcal D}[{\mathcal E}^{\alpha}] =\{u\in L^2({\mathbb R}^d; dx): {\mathcal E}^{\alpha}(u,u)<\infty\}\). The author shows that if the set \(\{x\in {\mathbb R}^d: \alpha(x)=-\infty\) or \(+\infty\}\) has Lebesgue measure zero, then \(({\mathcal E}^{\alpha},{\mathcal D}[{\mathcal E}^{\alpha}])\) is a Dirichlet form on \(L^2({\mathbb R}^d; dx)\) in the wide sense. Moreover, necessary and sufficient conditions are given in order that the family of all uniformly continuous functions on \({\mathbb R}^d\) with compact support be contained in \({\mathcal D}[{\mathcal E}^{\alpha}] \). In the second part of the paper the author studies path properties of the corresponding processes in the one-dimensional case. Specifically, conditions in terms of the variable index \(\alpha\) are given so that a point is exceptional. Finally, recurrence of the process is shown to depend on the behavior of \(\alpha\) at infinity.
scientific article; zbMATH DE number 1744767