Continuity and singularity of the intersection local time of stable processes in \({\mathbb{R}}^ 2\) (Q1102631)

Let \(X_ t\) be a planar stable process of index \(\beta\). \(\alpha\) is said to be an intersection local time of \(X_ t\) if \[ \int f(x)\alpha (x,B)dx=\iint_{B}f(X_ t-X_ S)dsdt \] for all bounded Borel functions f and sets \(B\subseteq R^ 2_+\). The author shows that if \(4/3<\beta <2\), then an intersection local time exists which is weakly continuous in \(x\neq 0\), while \[ \alpha (x,[0,T]^ 2)-cT/| x|^{2-\beta} \] has an extension continuous for all x, where c is a constant which depends upon \(\beta\).