Integral representation of renormalized self-intersection local times

Integral representation of renormalized self-intersection local times

In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the d-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1). As a consequence, we derive the existence of some exponential moments for this random variable.

Renormalized self-intersection local times and Wick power chaos processes

Sufficient conditions are obtained for the continuity of renormalized self-intersection local times for the multiple intersections of a large class of strongly symmetric Lévy processes in Rm, m = 1, 2. In R2 these include Brownian motion and stable processes of index greater than 3/2, as well as many processes in their domains of attraction. In R1 these include stable processes of index 3/4 < β...

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

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.

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...