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

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Let (Xₜ,t≥ 0) be a random walk on Zᵈ. Let lₜ(x)= ∫₀ᵗ δₓ(Xₛ)ds the local time at the state x and Iₜ= ∑ᵈ lₜ(x)q the q-fold self-intersection local time (SILT). In Castell Castell proves a large deviations principle for the SILT of the simple random walk in the critical case q(d-2)=d. In the supercritical case q(d-2)>d, Chen and Mörters obtain in ChenMorters a large deviations principle for the intersection of q independent random walks, and Asselah obtains in Asselah5 a large deviations principle for the SILT with q=2. We extend these results to an α-stable process (i.e. α∈]0,2]) in the case where q(d-α)≥ d.