Penalising symmetric stable Lévy paths

Abstract

Limit theorems for the normalized laws with respect to two kinds of weight functionals are studied for any symmetric stable Lévy process of index . The first kind is a function of the local time at the origin, and the second kind is the exponential of an occupation time integral. Special emphasis is put on the role played by a stable Lévy counterpart of the universal $\sigma$-finite measure, found in [9] and [10], which unifies the corresponding limit theorems in the Brownian setup for which $\alpha =2$.