Large deviations for self-intersection local times in subcritical dimensions

Large and moderate deviations for intersection local times

We study the large and moderate deviations for intersection local times generated by, respectively, independent Brownian local times and independent local times of symmetric random walks. Our result in the Brownian case generalizes the large deviation principle achieved in Mansmann (1991) for the L2-norm of Brownian local times, and coincides with the large deviation obtained by Csörgö, Shi and...

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 intersection local times in critical dimension

Let (X t , t ≥ 0) be a continuous time simple random walk on Z d , and let l T (x) be the time spent by (X t , t ≥ 0) on the site x up to time T. We prove a large deviations principle for the q-fold self-intersection local time I T = x∈Z d l T (x) q in the critical dimension d = 2q q−1. When q is integer, we obtain similar results for the intersection local times of q independent simple random ...

Self-intersection local times of random walks: exponential moments in subcritical dimensions

Fix p > 1, not necessarily integer, with p(d − 2) < d. We study the p-fold self-intersection local time of a simple random walk on the lattice Zd up to time t . This is the p-norm of the vector of the walker’s local times, t . We derive precise logarithmic asymptotics of the expectation of exp{θt‖ t‖p} for scales θt > 0 that are bounded from above, possibly tending to zero. The speed is identif...

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