Large deviations for self-intersection local times of stable random walks

Let (Xt,t≥0)(Xt,t≥0) be a random walk on ZdZd. Let lT(x)=∫0Tδx(Xs)ds be the local time at the state xx and IT=∑x∈ZdlT(x)qIT=∑x∈ZdlT(x)q the qq-fold self-intersection local time (SILT). In [5] Castell proves a large deviations principle for the SILT of the simple random walk in the critical case q(d−2)=dq(d−2)=d. In the supercritical case q(d−2)>dq(d−2)>d, Chen and Mörters obtain in [10] a large deviations principle for the intersection of qq independent random walks, and Asselah obtains in [1] a large deviations principle for the SILT with q=2q=2. We extend these results to an αα-stable process (i.e. α∈]0,2]α∈]0,2]) in the case where q(d−α)≥dq(d−α)≥d.