Large deviations for renormalized self-intersection local times of   stable processes
  We study large deviations for the renormalized self-intersection local time of d-dimensional stable processes of index \beta \in (2d/3,d]. We find a difference between the upper and lower tail. In addition, we find that the behavior of the lower tail depends critically on whether \beta <d or \beta =d. 
Richard Bass, Xia Chen and Jay Rosen