Large deviations for self-intersection local times in subcritical dimensions

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$bethelocaltimeatsite$x$and$ I_t= \sum\limits_{x\in$\mathbb{Z}$^d} l_t(x)^p $ the p-fold self-intersection local time (SILT). Becker and K"onig 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.