Self-intersection local time: Critical exponent, large deviations, and laws of the iterated logarithm

Abstract

If β_t is renormalized self-intersection local time for planar Brownian motion, we characterize when $\mathbb{E}e^{\gamma\beta_{1}}$ is finite or infinite in terms of the best constant of a Gagliardo–Nirenberg inequality. We prove large deviation estimates for β₁ and −β₁. We establish lim sup and lim inf laws of the iterated logarithm for β_t as t→∞.