In this paper, we study a renormalized self-intersection local time for superprocesses over stable processes and classical diffusions. When the renormalization breaks down, we obtain limit theorems.
"Renormalization and Limit Theorems for Self-Intersections of Superprocesses." Ann. Probab. 20 (3) 1341 - 1368, July, 1992. https://doi.org/10.1214/aop/1176989694