Exponential moments of self-intersection local times of stable random walks in subcritical dimensions

TL;DR

This paper derives precise asymptotic probabilities for the self-intersection local times of stable random walks in subcritical dimensions, extending previous theoretical results in the field.

Contribution

It provides new logarithmic asymptotics for the probability of large self-intersection local times in stable random walks, covering all scales beyond the expectation.

Findings

Established asymptotic formulas for $P(I_t \,\geq\, r_t)$

Extended previous results to non-integer $p$ and all scales $r_t$

Confirmed the subcritical dimension condition $p(d - \alpha) < d$

Abstract

Let $(X_t, t \geq 0)$ be an $\alpha$-stable random walk with values in $\Z^d$. Let $l_t(x) = \int_0^t \delta_x(X_s) ds$ be its local time. For $p>1$, not necessarily integer, $I_t = \sum_x l_t^p(x)$ is the so-called $p$-fold self- intersection local time of the random walk. When $p(d -\alpha) < d$, we derive precise logarithmic asymptotics of the probability $P(I_t \geq r_t)$ for all scales $r_t \gg \E(I_t)$. Our result extends previous works by Chen, Li and Rosen 2005, Becker and K\"onig 2010, and Laurent 2012.