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

Abstract

Let $(X_t,t\ge 0)$ be an $\alpha$-stable random walk with values in ${\mathbb{Z}}^d$. 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 $P(I_t\ge r_t)$ for all scales $r_t \gg \E(I_t)$. Our result extends previous works by Chen, Li and Rosen 2005, Becker and König 2010, and Laurent 2012.