Integral representation of renormalized self-intersection local times

Moderate deviations and laws of the iterated logarithm for the renormalized self-intersection local times of planar random walks

We study moderate deviations for the renormalized self-intersection local time of planar random walks. We also prove laws of the iterated logarithm for such local times

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 β ∈ (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 β < d or β = d.

Large Deviations for Renormalized Self-intersection Local times of Stable Processes by Richard Bass,1

Derivatives of self-intersection local times

We show that the renormalized self-intersection local time γt(x) for both the Brownian motion and symmetric stable process in R is differentiable in the spatial variable and that γ′ t(0) can be characterized as the continuous process of zero quadratic variation in the decomposition of a natural Dirichlet process. This Dirichlet process is the potential of a random Schwartz distribution. Analogo...

Markov Paths, Loops and Fields (preliminary Version)

We study the Poissonnian ensembles of Markov loops and the associated renormalized self intersection local times.