Large deviations for self-intersection local times in subcritical dimensions

Abstract

Let $(X_t,t\ge 0)$ be a random walk on ${\mathbb{Z}}^d$. Let $l_t(x)={\int }_0^t{\delta }_x(X_s)ds$ be the local time at site $x$ and $I_t=\sum _{x\in {\mathbb{Z}}^d}{l}_t(x{)}^p$ the p-fold self-intersection local time (SILT). Becker and König have recently proved a large deviations principle for $I_t$ for all $(p,d)\in {\mathbb{R}}^d\times {\mathbb{Z}}^d$ such that $p(d-2)<2$. We extend these results to a broader scale of deviations and to the whole subcritical domain $p(d-2)<d$. Moreover we unify the proofs of the large deviations principle using a method introduced by Castell for the critical case $p(d-2)=d$ and developed by Laurent for the critical and supercritical case $p(d-\alpha )\ge d$ of $\alpha$-stable random walk.