Large deviations and renormalization for Riesz potentials of stable intersection measures

Abstract

We study the object formally defined as (0.1) γ ( [ 0 , t ] 2 ) = ∬ [ 0 , t ] 2 | X s − X r | − σ d r d s − E ∬ [ 0 , t ] 2 | X s − X r | − σ d r d s , where X t denotes the symmetric stable processes of index 0 < β ≤ 2 in R d . When β ≤ σ < min { 3 2 β , d } , this has to be defined as a limit, in the spirit of renormalized self-intersection local time. We obtain results about the large deviations and laws of the iterated logarithm for γ . This is applied to obtain results about stable processes in random potentials.