Self-intersection local times of additive processes: Large deviation and law of the iterated logarithm

Abstract

Recently, we studied the large deviations for the local times of additive stable processes. In this work, we investigate the upper tail behaviors of the self-intersection local times of additive stable processes. Let X 1 ( t ) , … , X p ( t ) be independent, d -dimensional symmetric stable processes with stable index 0 < α ≤ 2 and consider the additive stable process X ¯ ( t 1 , … , t p ) = X 1 ( t 1 ) + ⋯ + X p ( t p ) . Under the condition d < α p , we compute large deviation probabilities for the self-intersection local time ∫ ∫ [ 0 , 1 ] p × [ 0 , 1 ] p δ 0 ( X ¯ ( r 1 , … , r p ) − X ¯ ( s 1 , … , s p ) ) d r 1 d s 1 ⋯ d r p d s p run by the multi-parameter field X ¯ ( t 1 , … , t p ) . Our theorem applies to the law of the iterated logarithm and our approach relies on Fourier analysis, moment computation, time exponentiation and some general methods developed along the lines of probability in Banach space.