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

Let (Xt, t >= 0) be an alpha-stable random walk with values in Z(d). Let l(t)(x) = integral(t)(0) delta(x) (Xs) ds be its local time. For p > 1, not necessarily integer, I-t = Sigma(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 >= r(t)] for all scales rt >> E[I-t]. Our result extends previous works by Chen, Li and Rosen [Electron. J. Probab. 10 (2005) 577-608], Becker and Konig [ Probab. Theory Related Fields 154 (2012) 585-605] and Laurent [Electron. J. Probab. 17 (2012) 1-20].