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

Abstract

Let ( X t , t ≥ 0 ) be a random walk on Z d . Let l T ( x ) = ∫ 0 T δ x ( X s ) d s be the local time at the state x and I T = ∑ x ∈ Z d l T ( x ) q the q -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 ) = d . In the supercritical case q ( d − 2 ) > d , Chen and Mörters obtain in [10] a large deviations principle for the intersection of q independent random walks, and Asselah obtains in [1] 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 .